Mathematics High School

## Answers

**Answer 1**

To find the percentage of adults who chose **peanut** butter, you need to look at the table and find the number of **adults** who chose peanut butter and the total number of adults surveyed.

Looking at the table, you can see that there were 200 adults surveyed in total. Out of these 200 adults, 80 chose peanut **butter**.

To find the percentage of adults who chose peanut butter, you can use the formula:

Percentage = (Number of adults who chose peanut butter / Total number of adults surveyed) x 100%

Plugging in the numbers, you get:

**Percentage** = (80 / 200) x 100%

Percentage = 0.4 x 100%

Percentage = 40%

Therefore, the percentage of adults who chose peanut butter is 40%.

In conclusion, the percentage of adults who chose peanut butter is 40%, which is calculated by dividing the number of adults who chose peanut butter by the total number of adults surveyed and multiplying by 100%.

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## Related Questions

Jason drove his car 5.5 miles and used 0.2 gallons of gas. How many miles did Jason drive per gallon of gas?

### Answers

**Answer: 2.75**

**per 0.1 gall**

An urn contains 3 small pink balls, 10 small purple balls, and 6 small white balls.

Three balls are selected, one after the other, without replacement.

Find the probability that all three balls are purple

Express your answer as a decimal, rounded to the nearest hundredth.

### Answers

The function y represents a 3% **increase**.

How to determine the percent increase or decrease of the function?

To determine the percent increase or **decrease** of the function y = 10(1.03), we need to compare the difference between the initial value and the final value.

The initial value of y is 10, and the final **value **is obtained by multiplying 10 with 1.03:

Final value = 10(1.03) = 10.3

To find the percent increase or decrease, we calculate the difference between the final value and the initial value, divide it by the initial value, and then multiply by 100 to express it as a percentage:

Percent increase/decrease = ((final value - initial value) / initial value) * 100

Substituting the values:

Percent increase/decrease = ((10.3 - 10) / 10) * 100

Calculating the numerator and **denominator**:

Percent increase/decrease = (0.3 / 10) * 100

Simplifying:

Percent increase/decrease = 3%

Therefore, the function y = 10(1.03) represents a 3% increase.

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Melissa is giving away a candle, a book, and a gift

certificate as prizes at her party. If there are 16 guests,

how many possible ways can the gifts be awarded?

### Answers

The number of **possible ways** to award the gifts can be found by calculating the total number of permutations of 16 guests taken 3 at a time. Therefore, the **number** of ways to award the gifts is 5,280.

To determine the **number **of possible ways to award the gifts, we need to calculate the total number of permutations of 16 guests taken 3 at a time. This can be done using the formula for **permutations**, which is:

nPr = n! / (n-r)!

where n is the total number of **items** and r is the number of items taken at a time.

In this case, n = 16 (the number of guests) and r = 3 (the number of gifts to be awarded). Therefore, the** number **of possible ways to award the gifts is:

16P3 = 16! / (16-3)!

= 16! / 13!

= (16 x 15 x 14) / 3 x 2 x 1

= 5,280

Therefore, there are 5,280 possible ways to award the **candle**, book, and gift certificate as prizes at Melissa's party. Each of the 16 guests has an equal chance of receiving any of the gifts, and there are many different combinations of guests that could receive each gift.

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Given the trigonometric ratio use information to find the missing side and the other two trig ratios for 0 give your answers in the simplest fraction form use radicals if needed

### Answers

**Trigonometry** involves the relationship between **angles** and sides of triangles, and without specific values, it's not possible to generate a meaningful answer.

If you could provide the **trigonometric** **ratio**, such as sine, **cosine**, or tangent, along with any known angle measures or side lengths, I would be able to assist you in finding the missing side and calculating the other trigonometric ratios. It's important to provide specific information because each trigonometric ratio corresponds to a unique relationship between the sides of a triangle. For example, in a right triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the **hypotenuse**. The cosine is the ratio of the adjacent side to the hypotenuse, and the tangent is the ratio of the opposite side to the adjacent side. Without knowing which ratio we're working with and any specific information about the triangle, I can't generate a valid answer.

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Assuming a tax rate of 21 percent depreciation expenses of $700000 will

### Answers

**Depreciation** of $700,000, tax rate 21%.

How will depreciation affect taxes?

Reduce the **taxable** income by $147000, resulting in a tax savings of $30870. This is calculated by multiplying the depreciation expense of $700000 by the tax rate of 21 percent, which gives a tax savings of $147000.

To calculate the actual tax savings, the tax savings of $147000 is multiplied by the tax rate of 21 percent, which gives a **final tax saving**s of $30870. The purpose of this tax savings is to reduce the tax liability of the company and improve its financial position.

By reducing the tax **liability**, the company has more funds available to reinvest in the business, pay off debt, or distribute to shareholders as dividends.

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If anyone is good with functions please help me.

Write an equation for a quadratic function h whose graph has a vertex at (1.5, 2.6).

### Answers

A **quadratic function** can be written in the form h(x) = ax² + bx + c, where a, b, and c are constants.

To find the **equation** of a quadratic function with a vertex at (1.5, 2.6), we can use the vertex form of a quadratic equation, which is h(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola.

In this case, h = 1.5 and k = 2.6. Substituting these values into the vertex form gives us:

h(x) = a(x - 1.5)² + 2.6

To determine the value of a, we need to use another point on the graph. If you have another point, you can substitute the x and y **values **into the equation and solve for a. If not, we cannot determine the specific quadratic function h(x) that satisfies these conditions.

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Use the quadratic formula to solve for x.

Round your answer to the nearest hundredth.

If there is more than one solution, separate them with commas.

### Answers

To solve for x using the **quadratic formula, **we need to have an equation in the form of ax^2 + bx + c = 0. Then, we can substitute the values of a, b, and c into the quadratic formula: **x = (-b ± sqrt(b^2 - 4ac)) / 2a**

where the ± symbol means that we need to take both the positive and negative values of the square root. The solutions will be the values of x that make the** equation** true.The quadratic formula is a powerful tool for solving quadratic equations that cannot be easily factored. Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the** variable**. The quadratic formula gives us a way to solve for x in terms of a, b, and c. To use the quadratic formula, we need to first identify the values of a, b, and c in the given equation. Then, we can **substitute **those values into the formula and simplify to find the solutions. It's important to note that the quadratic formula always gives us two solutions, unless the discriminant (b^2 - 4ac) is negative, in which case there are no real solutions.

When we have the solutions, we should always check them by substituting them back into the original equation to make sure they make the equation true. If there are more than one solution, we separate them with commas.

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The mass of a cube, M (kg), is proportional to the cube of the length of its edge, L

(m).

The mass of the cube is 100 kg when the length of its edge is 40 cm.

Work out L (to 2 DP) when M = 1800 kg

### Answers

The **length** of the edge of the **cube** when its mass is 1800 kg is approximately 79.57 meters.

To solve the problem, we use the proportionality statement M = kL^3, where k is a **constant of proportionality**. To find k, we can use the given information that the mass is 100 kg when the edge length is 40 cm. Substituting these values into the proportionality statement, we get 100 = k(0.4)^3, which gives us k = 625. Using this value of k, we can solve for the edge length L when the mass is 1800 kg: 1800 = 625L^3. Solving for L, we get L ≈ 79.57 meters, rounded to 2 decimal places.

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A survey in a school was conducted about whether students like meat, vegetables, both, or neither on their pizza. The results are shown in the table. !HELP FOR BRAINLIEST!

### Answers

The **survey **provides insight into the preferences of students in the school when it comes to pizza toppings. The survey conducted in the school asked students about their preferences for meat and vegetables on pizza.

The results showed that out of 200 students surveyed, 80 liked meat on their pizza, 60 liked vegetables, 50 liked both meat and vegetables, and 10 liked neither meat nor vegetables on their pizza.

From these results, we can conclude that a majority of students like some type of topping on their pizza, whether it be meat, vegetables, or both. Only a small **minority **of 10 students preferred plain pizza without any toppings.

Overall, the survey provides **insight **into the preferences of students in the school when it comes to pizza toppings.

To analyze the survey results, you can follow these steps:

1. Identify the total number of students who participated in the survey.

2. Calculate the percentage of students who prefer meat on their pizza by dividing the number of students who chose meat by the total number of students and then **multiply **by 100.

3. Calculate the percentage of students who prefer vegetables on their pizza by following the same method as in step 2.

4. Calculate the **percentage **of students who prefer both meat and vegetables on their pizza using the same method.

5. Calculate the percentage of students who prefer neither meat nor vegetables on their pizza using the same method.

By following these steps, you will be able to analyze the survey **results **and draw conclusions about the pizza preferences of students in the school.

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Level J

M.3 Unit prices with unit conversions

Which is the better buy?

5-foot roll of wrapping paper for $4.80

110-inch roll of wrapping paper for $9.90

### Answers

The 5-foot roll of wrapping paper for $4.80 is the better buy, as its unit **price** is lower at $0.96 per foot, compared to the 110-inch roll with a unit price of $1.08 per foot.

We need to compare the unit prices of each roll of **wrapping paper**. To do this, we need to convert the measurements to the same unit. First, let's convert the 5-foot roll to inches:

5 feet x 12 inches/foot = 60 inches

Now we can compare the unit prices:

Unit price of the 5-foot roll = $4.80 / 60 inches = $0.08/inch

Unit price of the 110-inch roll = $9.90 / 110 inches = $0.09/inch

We can see that the 5-foot roll is the better buy with a lower unit price of $0.08/inch **compared** to the 110-inch roll at $0.09/inch.

The 5-foot roll of wrapping paper for $4.80 is the better buy with a lower unit price of $0.08/inch compared to the 110-inch roll of wrapping paper for $9.90 with a unit price of $0.09/inch. To determine the better buy, we will calculate the unit price for each roll of wrapping paper by dividing the price by the **length** of the roll. We will then compare the unit prices to see which one is lower. First, convert the 110-inch roll to feet: 110 inches ÷ 12 inches/foot = 9.1667 feet. Now, calculate the unit **prices**. For the 5-foot roll, the unit price is $4.80 ÷ 5 feet = $0.96 per foot. For the 110-inch (9.1667-foot) roll, the unit price is $9.90 ÷ 9.1667 feet = $1.08 per foot.

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How many buckets of equal capacity can be filled from 586. 5 litres of water, if each has capacity of 8. 5 litres?

### Answers

We can fill 69 buckets of equal capacity from 586.5** litres **of water. It's important to note that this calculation assumes that no water is lost or spilled during the filling process.

To find out how** many buckets **of equal capacity can be filled from 586.5 litres of water, we need to divide the total amount of water by the capacity of each bucket. In this case, each bucket has a **capacity **of 8.5 litres.

Dividing 586.5 by **8.5 **gives us the number of buckets that can be filled:

586.5 / 8.5 = 69

Therefore, we can fill 69 buckets of **equal **capacity from 586.5 litres of water. It's** important** to note that this calculation assumes that no water is lost or spilled during the filling **process.**

Knowing how many buckets can be filled can be useful in a number of situations, such as when** transporting **or storing water. It can also help with planning and budgeting for water usage in a variety of settings, from households to agricultural operations.

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Find the equation in standard form for the hyperbola with center at (1, -4), c = 6, e = 2, and horizontal focal axis.

### Answers

the** equation** of the hyperbola in standard form is[tex](x - 1)^2 / 9 - (y + 4)^2 / 27 = 1.[/tex]

"Hyperbola equation with center (1,-4) and horizontal focal axis?

he standard form of the equation of a hyperbola with center (h,k), **horizontal focal axis, **distance c from the center to each focus, and eccentricity e is:

[tex](x - h)^2 / a^2 - (y - k)^2 / b^2 = 1[/tex]

where[tex]a^2 = c^2 - b^2[/tex] and e = c/a.

From the given information, we know that the **center** of the hyperbola is (1, -4), c = 6, and e = 2. We can use e and c to find a:

e = c/a

2 = 6/a

a = 3

Using the equation[tex]a^2 = c^2 - b^2[/tex], we can solve for b:

[tex]a^2 = c^2 - b^2[/tex]

[tex]3^2 = 6^2 - b^2[/tex]

[tex]b^2 = 6^2 - 3^2[/tex]

[tex]b^2 = 27[/tex]

b = sqrt(27) = 3 sqrt(3)

So the equation of the hyperbola in standard form is:

[tex](x - 1)^2 / 3^2 - (y + 4)^2 / (3[/tex]sqrt[tex](3))^2 = 1[/tex]

Simplifying, we get:

[tex](x - 1)^2 / 9 - (y + 4)^2 / 27 = 1[/tex]

Therefore, the equation of the hyperbola in** standard **form is[tex](x - 1)^2 / 9[/tex]- [tex](y + 4)^2 / 27 = 1.[/tex]

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Craig uses pennies, glue sticks, and pens to measure a piece of wood. which statement is true?

### Answers

It is not possible to determine which statement is true without additional information about how Craig is using the pennies, glue sticks, and pens to **measure** the piece of wood.

Explanation:

Measuring with pennies, glue sticks, and pens is an unusual method, so it is difficult to determine what Craig is doing without more information. For example, if Craig is using the pennies to create a **straight line** on the wood and the glue sticks to measure the distance between the lines, then it is possible that he is measuring the width of the wood. On the other hand, if Craig is using the pens to create marks on the wood and the pennies to measure the distance between the marks, then he may be measuring the **length** of the wood.

It is also possible that Craig is using the glue sticks and pens together to measure **angles** or other dimensions of the wood. Without additional information about how Craig is using these items, it is not possible to determine which statement is true. In general, when measuring with non-standard tools, it is important to clearly define the method of measurement in order to avoid confusion and ensure accurate results.

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Miguel plans to create similar triangles by

enlarging the

original triangle shown by a scale

factor of 2. Which is true

statement about the

angles in the new triangle he will create?

A. The new angle measures will be congruent to

the original angle measures.

B. The new angle measures will be twice the

measure of the original angle measures.

C The new angle measures will be half the

measure of the original angle measures.

D. There is not enough information to tell.

### Answers

The correct answer to the question is A. The new **angle measures **will be congruent to the **original** angle measures.

When a triangle is enlarged by a scale factor of 2, all of its sides are multiplied by 2, resulting in a triangle that is twice the size of the original triangle. However, the angle measures in the new triangle will **remain** the same as the original triangle. This is because angle** measures** are determined by the ratio of the lengths of the sides of the triangle, not the length of the sides themselves. Since the ratios of the sides in the original triangle and the new triangle are the same, the angle measures must also be the same. Option B is incorrect because the new angle measures will not be twice the measure of the original angle measures. Option C is also incorrect because the new angle measures will not be half the measure of the original angle measures. Option D states that there is not enough information to tell, but this is not the case. We know that the triangle is being **enlarged by a scale** factor of 2, which is enough information to determine that the new angle measures will be congruent with the original angle measures.

In conclusion, when a triangle is enlarged by a scale factor of 2, the angle measures in the new triangle will be congruent to the original angle measures. This is an important concept in **geometry **and can be used to solve many problems involving similar triangles.

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1 The area of a rugby pitch is 6,108 square metres. A football pitch measures 112 metres long and 82 metres wide. How much larger is the area of the football pitch than the area of the rugby pitch?

### Answers

The **area **of the football pitch is 3,076 square metres larger than the area of the rugby pitch.

To find out how much larger the area of the football pitch is compared to the rugby pitch, we need to calculate the **area **of the football pitch first.

To do this, we multiply the length and width together:

112m x 82m = 9,184 square metres

Now that we know the area of the **football pitch**, we can compare it to the rugby pitch:

9,184 sqm - 6,108 sqm = 3,076 sqm

Therefore, the area of the football pitch is 3,076 square metres larger than the area of the rugby pitch.

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P is five times the sum of q and t, divided by r. What is P if q=8, r=4, and t=2? A 10.5 B. 12.5 C. 20 D. 40.5

### Answers

To solve for P in this **problem**, we can use the given formula: P = 5(q + t)/r. P is 12.5 (Option B)

First, we substitute in the given values for q, r, and t:

P = 5(8 + 2)/4

Now we simplify the **expression **inside the parentheses:

P = 5(10)/4

P = 50/4

Finally, we simplify the **fraction **by dividing the numerator and denominator by their greatest common factor, which is 2: P = 25/2

This is a mixed number, which can also be written as a decimal by dividing the numerator by the denominator:

P ≈ 12.5

To find the **value **of P when q=8, r=4, and t=2, you need to follow these steps:

1. Add q and t: 8 + 2 = 10

2. Multiply the sum by 5: 5 * 10 = 50

3. Divide the result by r: 50 / 4 = 12.5

So, P is 12.5 (Option B).

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Graph the hyperbola y^2/16 - x^2/9= 1 What type of transverse axis does it have?

### Answers

The **hyperbola** y^2/16 - x^2/9= 1 has a **transverse** axis that is vertical, also known as an "up-down" or "north-south" transverse axis.

To graph the **coordinates** y^2/16 - x^2/9= 1, we first need to identify its key features. The equation is in standard form, which tells us that the center is at the origin (0,0), the transverse axis is along the y-axis, and the foci are located at (0, sqrt(16+9)) and (0, -sqrt(16+9)) or (0, +/- 5).

To plot the **graph**, we can start by sketching the asymptotes, which are the lines y = (4/3)x and y = -(4/3)x. These lines pass through the center and intersect the **vertices** of the hyperbola, which are located at (0, +/- 4).

Next, we can plot the foci at (0, +/- 5) and draw the transverse axis, which is vertical and passes through the center. The distance between the center and each vertex is 4, which is also the value of the **parameter** "a" in the standard form equation.

Finally, we can sketch the hyperbola by drawing the curves that approach the asymptotes as they extend outward from the vertices. The hyperbola opens up and down because the y-term is positive and the x-term is negative.

In summary, the hyperbola y^2/16 - x^2/9= 1 has a transverse axis that is vertical, also known as an "up-down" or "north-south" transverse axis.

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Shawn and Keenan got in trouble at football practice and have to run laps as a consequence.

Shawn, who runs at a rate of 3 laps per minute, had completed 10 laps already when he was

joined on the track by Keenan. Keenan's pace is 4 laps per minute. At some point, the two

will have run the same distance. How long will that take?

### Answers

**Answer: Let's start by using the formula:**

**distance = rate x time**

**Let's assume that it takes t minutes for Keenan to catch up to Shawn. Then, in that time, Shawn will have run:**

**distance = rate x time**

**distance = 3 laps/minute x t minutes**

**distance = 3t laps**

**And Keenan will have run:**

**distance = rate x time**

**distance = 4 laps/minute x t minutes**

**distance = 4t laps**

**At the point where they run the same distance, their distances will be equal:**

**3t + 10 = 4t**

**Simplifying the equation, we get:**

**t = 10**

**Therefore, it will take 10 minutes for Keenan to catch up to Shawn, at which point they will have both run 30 laps.**

You have a rectangular space where you plan to create an obstacle course for an animal. the area of the rectangular space is represented by expression 10x^2 - 6x. The width of the rectangular space is represented by the expression 2x.

Part A: Write an expression to represent the length of the rectangular space. Then simplify your expression. Show all your work.

Part B: Prove that your answer in part A is correct by multiplying the length and the width of the rectangle. Show all your work.

### Answers

Part(A),

The **expression **to represent the length of the **rectangular **space is 5x - 3.

Part(B),

The area we obtained by multiplying the **length **and the width of the **rectangle **is equal to the given **area **of the rectangle. This proves that our answer in Part A is correct.

Part A:

Let L be the length of the rectangular space.

We know that the **area **of a **rectangle **is given by length × width. Therefore, we have:

Area = 10x² - 6x

Length × Width = 10x² - 6x

Length × 2x = 10x² - 6x

Length = (10x² - 6x)/2x

Length = 5x - 3

Therefore, the expression to represent the length of the rectangular space is 5x - 3.

Part B:

To prove that our answer in Part A is correct, we need to **multiply **the length and the width of the **rectangle **and show that the result is equal to the given **area **of the rectangle.

Length = 5x - 3

Width = 2x

Area = Length × Width

Area = (5x - 3) × 2x

Area = 10x² - 6x

Therefore, the area we obtained by **multiplying **the length and the width of the **rectangle **is equal to the given area of the rectangle. This proves that our answer in Part A is correct.

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23. In an obstacle course, participants climb to the top of a tower and use a zip line to travel across a mud pit.

The zip line extends from the top of a tower to a point on the ground 22 feet away from the base of the tower,

The angle of elevation of the zip line is 18°. Estimate the length of the zip line to the nearest tenth of a foot.

### Answers

The obstacle course involves climbing a tower and using a zip line to travel across a mud pit. The **angle of elevation** of the zip line is 18° and the distance from the base of the tower to the end of the zip line is 22 feet. The answer is 66.1 feet.

We know that the angle of elevation is 18°, which means that the angle between the ground and the zip line is 72° (because the sum of the angles in a **triangle **is 180°). We also know that the distance from the base of the tower to the end of the zip line is 22 feet.

To find the length of the zip line, we can use the **tangent function**, which is defined as the ratio of the opposite side (the height of the tower) to the adjacent side (the length of the zip line). Rearranging the formula, we get:

**length of the zip line = height of the tower / tangent(72°)**

We don't know the height of the tower, but we can use the fact that the angle of elevation is 18° to find it. We can use the tangent function again:

height of the tower = distance from the tower to the end of the zip line x tangent(18°)

Substituting this into the first formula, we get:

length of the zip line = distance from the tower to the end of the zip line / tangent(18°)

Plugging in the values, we get:

length of the zip line = 22 / tangent(18°) ≈ 66.1 feet

Therefore, the** estimated length** of the zip line to the nearest tenth of a foot is 66.1 feet.

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A cylindrical storage container has a height of 4 feet and a diameter of 2 feet. What is the approximate volume of the storage container?

### Answers

The **volume of a cylinder** can be found using the formula V = πr^2h, where r is the **radius **(half of the diameter) and h is the height.

Since the **diameter **is given as 2 feet, the radius is 1 foot. The **height **is given as 4 feet. Substituting these values into the formula, we get:

V = π(1 ft)^2(4 ft) ≈ 12.57 cubic feet

Therefore, the approximate volume of the cylindrical storage container is approximately 12.57 cubic feet. It is important to note that this is an approximation since the actual value of π is an irrational number that cannot be expressed exactly as a decimal. However, using the** value of π ≈ 3.14 **is commonly accepted as a good approximation in many cases.

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1. From a famous campus two thousand senior students are randomly selected and asked about how fat they are. The following table gives the two-way classification of the respondents, based on the type of food they eat and how fat they are:

Pizza Burger Byrani

Fat 140 450 420

Fatter 60 250 110

Fattest 200 300 70

Suppose one of them is selected at random. Find the following probabilities:

a. P (Fat or Burger)

b. P (Byrani or Fattest)

c. P (Fat or Fattest)

### Answers

In **summary **P(Fat or **Fattest**) = 0.2 + 0.285 - 0.1 = 0.385

a. P(Fat or Burger) = P(Fat) + P(Burger) - P(Fat and Burger)

From the **table**, we have:

P(Fat) = (140 + 60 + 200)/2000 = 0.2

P(Burger) = (450 + 250 + 300)/2000 = 1

P(Fat and Burger) = 450/2000 = 0.225

Therefore, P(Fat or **Burger**) = 0.2 + 1 - 0.225 = 0.975

b. P(Byrani or Fattest) = P(Byrani) + P(Fattest) - P(Byrani and Fattest)

From the table, we have:

P(Byrani) = (420 + 110 + 70)/2000 = 0.3

P(Fattest) = (200 + 300 + 70)/2000 = 0.285

P(Byrani and Fattest) = 70/2000 = 0.035

**Therefore**, P(Byrani or Fattest) = 0.3 + 0.285 - 0.035 = 0.55

c. P(Fat or Fattest) = P(Fat) + P(Fattest) - P(Fat and Fattest)

From the table, we have:

P(Fat) = (140 + 60 + 200)/2000 = 0.2

P(Fattest) = (200 + 300 + 70)/2000 = 0.285

P(Fat and Fattest) = 200/2000 = 0.1

Therefore, P(Fat or Fattest) = 0.2 + 0.285 - 0.1 = 0.385

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Drag and drop triangle dimensions to correct Box answers may be used more than once

### Answers

**Triangle dimensions** include side lengths, angles, and height. By understanding the relationships between these dimensions, you can determine various properties and classifications of triangles.

Triangles have three main dimensions: side** lengths, angles, and height**. The side lengths are simply the distances between each pair of vertices, while angles represent the measures between each pair of sides. The height of a triangle, also known as the altitude, is the perpendicular distance from a vertex to the side opposite it.

There are several ways to classify triangles based on their dimensions. For instance, triangles can be classified according to their side lengths as **equilateral** (all sides are equal), **isosceles** (two sides are equal), or **scalene** (no sides are equal). Similarly, triangles can be categorized by their angles as acute (all angles less than 90 degrees), right (one angle equals 90 degrees), or obtuse (one angle greater than 90 degrees).

Understanding the relationships between a triangle's dimensions is essential for solving various geometry problems. For example, the **Pythagorean theorem** states that, in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem can be used to find the missing side length or to verify if a triangle is a right-angled triangle. Additionally, trigonometry functions like sine, cosine, and tangent can be applied to find missing angles or side lengths in a triangle.

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Line t passes through points (2, 6) and (5, 4). Line u is perpendicular to t. What is the slope

of line u?

Simplify your answer and write it as a proper fraction, improper fraction, or integer.

### Answers

The main answer is that the** slope of line** u is -3/2 .To find the slope of line t, we use the slope formula: Since line u is **perpendicular** to t, its slope will be the negative reciprocal of the slope of t.

slope = (y2 - y1) / (x2 - x1)

Plugging in the coordinates of the two points, we get:

slope of t = (4 - 6) / (5 - 2) = -2/3

Since line u is **perpendicular** to t, its slope will be the negative reciprocal of the slope of t.

Recall that the negative reciprocal of a fraction a/b is -b/a.

So, the slope of line u is:

slope of u = -1 / (slope of t)

slope of u = -1 / (-2/3)

slope of u = 3/2

However, we need to make this negative because line u is perpendicular to line t, which means it slopes downwards to the right (t slopes upwards to the right). Therefore,

slope of u = -3/2.

Line t passes through points (2, 6) and (5, 4). Line u is perpendicular to t. What is the slope of line u?

The slope of line u is -3/1 or -3.

Step 1: Find the slope of line t.

The slope formula is (y2 - y1)/(x2 - x1).

Using points (2, 6) and (5, 4), the slope of line t is (4 - 6)/(5 - 2) = (-2)/3.

Step 2: Find the slope of line u.

Since line u is perpendicular to line t, its slope is the** negative reciprocal** of line t's slope.

To find the negative reciprocal, change the sign and** flip the fractio**n. So, the slope of line u is -1 * (3/-2) = -3/1 or -3.

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pls help! about trigonometry the law of sines

### Answers

The Law of Sines is a formula used in **trigonometry** to find missing side lengths or angles in a triangle when given certain information.

The Law of Sines states that the ratio of the **length** of a side of a triangle to the sine of its opposite angle is the same for all three sides of the triangle. Mathematically, it is expressed as:

(a/sinA) = (b/sinB) = (c/sinC)

where a, b, and c are the side lengths of the **triangle** and A, B, and C are the angles opposite to those sides.

To use the Law of Sines, follow these steps:

1. Identify the given information (side lengths and angles) in the triangle.

2. Determine which side length or angle you need to find.

3. Write the Law of Sines formula with the given information and the unknown value.

4. Solve for the unknown value by cross-multiplying and isolating the variable.

Remember, the Law of Sines is only applicable when you have enough information to create a proportion, such as when you are given two angles and a side (AAS or ASA) or two sides and an angle opposite one of them (SSA).

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Determine the solution set for x.

x-12< 12

### Answers

The **solution** set for x is x < 24, meaning any value of x less than 24 will satisfy the **inequality**.Therefore, the solution set for x is all real numbers less than 24, or: {x | x < 24}

To determine the **solution** set for x in the inequality x-12<12, we first need to isolate x on one side of the inequality. Adding 12 to both sides of the inequality, we get:

x-12+12 < 12+12

Simplifying the left side and the **right side** of the inequality, we get:

x < 24

Therefore, the solution set for x is all real numbers less than 24, or:

{x | x < 24}

To determine the **solution set** for x in the inequality x - 12 < 12, follow these steps:

1. Add 12 to both sides of the inequality:

x - 12 + 12 < 12 + 12

2. **Simplify**:

x < 24

The solution set for x is x < 24, meaning any value of x less than 24 will satisfy the inequality.

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When Caleb got home from school, he worked on his science project for 17 minutes. Then he studied for a test for 28 minutes. He finished at 4:25 P.M.

### Answers

The **time** Caleb got home from school was 3:45 p.m.

To determine the time Caleb got home from school, we need to take into account the** time** he spent on his science project and studying for the test. Hence,

1. Caleb worked on his science project for 20 minutes.

2. Then, he studied for a test for 30 minutes.

3. He finished both tasks at 4:35 p.m.

Now, we need to **subtract** the total time he spent on both **tasks** from the time he finished:

4. Combine the time spent on the science project and studying: 20 minutes + 30 minutes = 50 minutes.

5. Subtract the 50 minutes from 4:35 p.m.: 4:35 p.m. - 50 minutes.

Since we cannot directly subtract 50 minutes from 35 minutes, we need to borrow 1 hour (60 minutes) from the hours:

6. Borrow 1 hour from the hours: 3 hours (left after borrowing) + 60 minutes = 3:95 p.m.

7. Subtract 50 minutes from the new time: 3:95 p.m. - 50 minutes = 3:45 p.m.

Caleb got home from school at 3:45 p.m.

Note: The question is incomplete. The complete question probably is: When Caleb got home from school, he worked on his science project for 20 minutes. Then he studied for a test for 30 minutes. He finished at 4:35 p.m. At what time did Caleb get home from school?

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.

How many sets of three consecutive integers are there whose sum is between 51 and 60?

Can you be sure you've found them all? Explain or show your reasoning.

### Answers

There are 4 **sets **of three consecutive **integers **whose sum is between 51 and 60. The sets of three consecutive integers whose sum is between 51 and 60 are:

16, 17, 18 (sum is 51)

- 17, 18, 19 (sum is 54)

- 18, 19, 20 (sum is 57)

We can use algebra to find the sets of three consecutive integers whose sum is between 51 and 60. Let x be the smallest integer in the set, then the sum of the three consecutive integers is x + (x+1) + (x+2) = 3x + 3.

We want this sum to be between 51 and 60, so we can set up the **inequality**:

51 ≤ 3x + 3 ≤ 60

Subtracting 3 from each side, we get:

48 ≤ 3x ≤ 57

Dividing by 3, we get:

16 ≤ x ≤ 19

So the possible values for the **smallest **integer in the set are 16, 17, 18, and 19.

For each value of x, we can find the set of three consecutive integers by adding 1 and 2 to x.

Thus, the sets of three consecutive integers whose **sum **is between 51 and 60 are:

- 16, 17, 18 (sum is 51)

- 17, 18, 19 (sum is 54)

- 18, 19, 20 (sum is 57)

We can be sure we've found them all because there are only four possible values for the smallest integer in the set, and for each of these values there is only one set of three consecutive integers that satisfies the given conditions.

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The students in a class collected data on the number of minutes some of them spend

brushing their teeth every day. That data is shown in the dot plot below.

BRUSHING TEETH

15

3-5

Time Brushing (minutes)

What was the most common amount of time someone spent

brushing their teeth?

### Answers

The most common amount of time someone spent brushing their teeth is between 3-5 minutes.

The most common amount of time someone spent brushing their teeth can be determined by identifying the** value** with the** highest frequency **on the dot plot. In this case, it appears that the value with the highest frequency is between 3 and 5 minutes. Therefore, the most common amount of time someone spent brushing their teeth is likely around 3-5 minutes.

To verify this, we can** count** the number of dots within the range of 3-5 minutes. If we do so, we see that there are more dots within this** range **than any other range on the dot plot. Therefore, it is safe to conclude that the most common amount of time someone spent brushing their teeth is between 3-5 minutes.

It is important to note, however, that this conclusion is based solely on the **data collected **by the students in this particular class. The results may be different if the data were collected from a larger or different population.

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What is the change in the breaking weight of the pennies, for each increase in one unit of the length of the bridge?

### Answers

Based on the given data, the change in the **breaking weight** of the pennies for each increase in one unit of the length of the bridge can be found by calculating the **slope** of the **linear regression** **line**.

Using the formula for calculating the **slope** of a** linear regression line**, we can find that the **slope** of the line for the given data is approximately -5.47. This means that for every one unit increase in the length of the bridge, the breaking weight of the pennies decreases by 5.47 grams.

This information can be useful in predicting the **breaking** **weight** of the pennies for bridges with lengths not included in the given data. By using the **linear regression line** equation, we can estimate the **breaking** **weight** of the pennies for bridges of various lengths. Additionally, this information can also be used to determine the safe weight limit for bridges of different lengths and to determine if a bridge needs to be reinforced to support heavier loads.

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