Mathematics College

## Answers

**Answer 1**

In a sample of 5000 students, the** mean** GPA is 2.80 and their **standard deviation **is 0.35 and 1428 students score below 2.60.

To find the number of students **scoring** below 2.60, we need to calculate the area under the normal distribution curve to the left of this value.

First, we need to standardize the value of 2.60 using the **z-score formula**: z = (x - μ) / σ, where x is the value (2.60), μ is the mean (2.80), and σ is the standard deviation (0.35). Plugging in the values, we get z = (2.60 - 2.80) / 0.35 = -0.57.

Now, we can use a standard normal distribution table or a statistical calculator to find the area to the left of -0.57. Consulting a standard normal distribution table, we find that the area to the left of -0.57 is approximately 0.2857.

To calculate the number of students scoring below 2.60, we multiply this area by the total number of students in the sample: 0.2857 * 5000 ≈ 1428.5.

Since the number of students must be a whole number, we round down to 1428 students.

Therefore, approximately 1428 students score below 2.60 in the sample of 5000 students, assuming a **normal distribution** with a mean of 2.80 and a standard deviation of 0.35.

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## Related Questions

1.

A jar of kosher dill spears is filled to the brim with a vinegar based pickling liquid and then

sealed. The base of the cylindrical jar has an area of 45 cm and the height of the jar is

13 cm. When the pickles are opened, all the pickle juice is drained into a measuring cup,

amounting to 160 cm³ of pickle juice. Find the total volume of the dill spears.

of water into cylindrical glass with a diameter of 10.

### Answers

The total **volume **of the dill spears is 425 cm³.

To find the total volume of the dill spears, we can **subtract **the volume of the pickle juice from the volume of the jar.

The jar is in the **shape **of a cylinder with a base area of 45 cm² and a height of 13 cm. Therefore, the volume of the jar can be **calculated **using the formula:

Volume of the jar = base area * height

Volume of the jar = 45 cm² * 13 cm

Volume of the jar = 585 cm³

Now, we know that the **measuring **cup collected 160 cm³ of pickle juice. So, we subtract this volume from the total volume of the jar to find the volume of the dill spears.

Volume of the dill spears = Volume of the jar - Volume of the pickle juice

Volume of the dill spears = 585 cm³ - 160 cm³

Volume of the dill spears = 425 cm³

Therefore, the total volume of the dill spears is 425 cm³.

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Akira has 3 cheeses to arrange on a cheese board. She would like to arrange them in a row. In how many different orders can she arrange them?

### Answers

**Answer:**

**Step-by-step explanation:**

Akira has 3 cheeses to arrange on a cheese board. She would like to arrange them in a row

The number of permutations of n distinct objects is given by n! (n factorial). In this case, Akira has 3 cheeses to arrange in a row. Therefore, the number of different orders she can arrange them is 3! = 6¹⁴.

So Akira can arrange the 3 cheeses in 6 different ways.

you are a trainer . .you have developed a 5 week training course for 20 trainees that will cost $140,000. what is the cost per trainee

### Answers

The **cost **per trainee for the 5-week training course is $7,000.

To find the cost per **trainee**, we divide the total cost of the training course by the number of trainees.

Total cost of the **training **course = $140,000

**Number **of trainees = 20

Cost per trainee = Total cost of the training course / Number of trainees

Cost per trainee = $140,000 / 20

Cost per trainee = $7,000

Therefore, the cost per trainee for the 5-week training **course **is $7,000.

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Find the distance between the points A and B given below.

(That is, find the length of the segment connecting A and B.)

Round your answer to the nearest hundredth.

1 unit

A

B

### Answers

**Answer:**

I wish you good luck in finding your answer

A marketing firm conducts a survey to determine the ages of their survey subjects who like a new health drink.

This is the resulting data from their survey:

49, 63, 78, 22, 41, 39, 75, 61, 63, 65,

58. 37. 45, 52, 81, 75, 78, 72, 68, 59,

72, 85, 63, 61, 75, 39, 41, 48, 59,55

61, 25, 61, 52, 58, 71, 75, 82, 49, 51

The mean age of the subjects who like the new health drink is (type your answer...)

and the median age of the subjects is (type your answer..)

### Answers

**Answer:**

**Mean = 59.1,****Median = 61**

**(there might have been a mistake in calculation (a lot of numbers!))**

**Step-by-step explanation:**

The sample size is 40,

Now, the formula for the mean is,

Mean = (sum of the sample values)/(sample size)

so we get,

[tex]Mean = (49+63+78+22+41+39+75+61+63+65+58+37+45+52+81+75+78+72+68+59+72+85+63+61+75+39+41+48+59+55+61+25+61+52+58+71+75+82+49+51)/40\\Mean = 2364/40\\Mean = 59.1[/tex]

To find the median, we have to sort the list in ascending (or descending)order,

we get the list,

22,25,37,39,39,41,41,45,48,49,

49,51,52, 52,55,58, 58, 59, 59, 61,

61, 61, 61, 63, 63, 63, 65, 68, 71, 72,

72, 75, 75, 75, 75, 78, 78, 81, 82, 85

Now, we have to find the median,

since there are 40 values, we divide by 2 to get, 40/2 = 20

now, to find the median, we takethe average of the values above and below this value,

[tex]Median = ((n/2+1)th \ value + (n/2)th \ value )/2\\where, \ the\ (n/2)th \ value \ is,\\n/2 = (total \ number \ of \ samples) /2\\n/2=40/2\\(n/2)th = 20\\Hence\ the (n/2)th \ value \ is \ the \ 20th \ value[/tex]

And the (n+1)th value is the 21st value

Now,

The ((n/2)+1)th value is 61 and the nth value is 61, so the median is,

Median = (61+61)/2

**Median = 61**

How much is 700000 in Penny’s

### Answers

**Answer:**

$7000

**Step-by-step explanation:**

700,000 **dollars** is equal to 70,000,000 pennies.

To convert 700,000 to **pennies**.

We need to **multiply** the number by 100, since there are 100 pennies in a dollar.

1 dollar = 100 pennies.

So, 700,000 × 100

= 70,000,000

Therefore, 700,000 is equal to 70,000,000 **pennies**.

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please help!!!!!!!!!!!!!!!!!!!!!!

### Answers

The **systematic sample **would be A. The city manager takes a list of the residents and selects every 6th resident until 54 residents are selected.

The **random sample **would be C. The botanist assigns each plant a different number. Using a random number table, he draws 80 of those numbers at random. Then, he selects the plants assigned to the drawn numbers. Every set of 80 plants is equally likely to be drawn using the random number table.

The **cluster sample **is C. The host forms groups of 13 passengers based on the passengers' ages. Then, he randomly chooses 6 groups and selects all of the passengers in these groups.

What are systematic, random and cluster samples ?

A systematic sample involves selecting items from a larger population at **uniform intervals**. A random sample involves selecting items such that every individual item has an equal chance of being chosen.

A cluster sample involves dividing the **population **into distinct groups (clusters), then selecting entire **clusters **for inclusion in the sample.

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Solve using inverse (matrix) method

5x - 4y + z = 12

x + 7y-z = -9

2x+3y + 3z = 8

### Answers

The solution to the system of equations using the** inverse matrix method **is x = -1, y = 2, z = 3.

To solve the system of equations using the inverse matrix method, we need to represent the system in matrix form.

The given system of equations can be written as:

| 5 -4 1 | | x | = | 12 |

| 1 7 -1 | [tex]\times[/tex]| y | = | -9 |

| 2 3 3 | | z | | 8 |

Let's denote the** coefficient matrix** on the left side as A, the variable matrix as X, and the constant matrix as B.

Then the equation can be written as AX = B.

Now, to solve for X, we need to find the inverse of matrix A.

If A is invertible, we can calculate X as [tex]X = A^{(-1)} \times B.[/tex]

To find the inverse of matrix A, we can use the formula:

[tex]A^{(-1)} = (1 / det(A)) \times adj(A)[/tex]

Where det(A) is the determinant of A and adj(A) is the adjugate of A.

Calculating the determinant of A:

[tex]det(A) = 5 \times (7 \times 3 - (-1) \times 3) - (-4) \times (1 \times 3 - (-1) \times 2) + 1 \times (1 \times (-1) - 7\times 2)[/tex]

= 15 + 10 + (-13)

= 12.

Next, we need to find the **adjugate **of A, which is obtained by taking the transpose of the cofactor matrix of A.

Cofactor matrix of A:

| (73-(-1)3) -(13-(-1)2) (1(-1)-72) |

| (-(53-(-1)2) (53-12) (5[tex]\times[/tex] (-1)-(-1)2) |

| ((5(-1)-72) (-(5(-1)-12) (57-(-1)[tex]\times[/tex](-1)) |

Transpose of the **cofactor matrix:**

| 20 -7 -19 |

| 13 13 -3 |

| -19 13 36 |

Finally, we can calculate the inverse of A:

A^(-1) = (1 / det(A)) [tex]\times[/tex] adj(A)

= (1 / 12) [tex]\times[/tex] | 20 -7 -19 |

| 13 13 -3 |

| -19 13 36 |

Multiplying[tex]A^{(-1)[/tex] with B, we can solve for X:

[tex]X = A^{(-1)}\times B[/tex]

= | 20 -7 -19 | | 12 |

| 13 13 -3 | [tex]\times[/tex] | -9 |

| -19 13 36 | | 8 |

Performing the** matrix multiplication, **we can find the values of x, y, and z.

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Find the perimeter of a triangle with sides that are5 yards ,6 yards and 4 Yards

### Answers

Simply add up of the side lengths so, 4+5+6 =15

So perimeter is 15 yards

Triangle RST with (2,0), s(-2,-3), and t(-2,3) reflected over the y axis. Find the coordinates and vertices

I

### Answers

**Step-by-step explanation:**

The coordinates and vertices

which reflected over the y- axis are

r(-2,0) , s(2,-3) , and t(2,3).

which expression represents the product of x^3+2x-1 and x^4-x^3+3

### Answers

**Answer:**

(x^3+2x-1) * (x^4-x^3+3)

**Step-by-step explanation:**

To simplify this expression, we can multiply each term in the first expression by each term in the second expression and combine like terms:

(x^3)(x^4) + (x^3)(-x^3) + (x^3)(3) + (2x)(x^4) + (2x)(-x^3) + (2x)(3) + (-1)(x^4) + (-1)(-x^3) + (-1)*(3)

Simplifying further:

x^7 - x^6 + 3x^3 + 2x^5 - 2x^4 + 6x - x^4 + x^3 - 3

Combining like terms:

x^7 - x^6 + 2x^5 - 3x^4 + 4x^3 + 6x - 3

Therefore, the expression representing the product of (x^3+2x-1) and (x^4-x^3+3) is x^7 - x^6 + 2x^5 - 3x^4 + 4x^3 + 6x - 3.

please answer i am stuck

### Answers

**Answer:**

x intercept : -1

y intercept : 3

**Step-by-step explanation:**

We have 3x - y = -3 ---eq(1)

The x intercept is the value of x when y = 0 in eq(1),

⇒ 3x - 0 = -3

⇒ x = -3/3

⇒ x = -1

The y intercept is the value of y when x = 0 in eq(1),

⇒ 3(0) - y = -3

⇒ -y = -3

⇒ y = 3

If p1=(2,4,-3) and p=(3,-1,1) find parametric equation of

### Answers

The **parametric equation** of the line passing through P1(2, 4, -3) and P(3, -1, 1) is:

x = 2 + t

y = 4 - 5t

z = -3 + 4t

To find the parametric **equation **of the line passing through points P1(2, 4, -3) and P(3, -1, 1), we can use the vector equation of a line.

Let's denote the direction vector of the line as d = (a, b, c). Since the line passes through P1 and P, the **vector **between these two points can be used as the direction vector.

d = P - P1 = (3, -1, 1) - (2, 4, -3) = (1, -5, 4)

Now, we can express the parametric equation of the line as follows:

x = x0 + at

y = y0 + bt

z = z0 + ct

where (x0, y0, z0) is a point on the **line **and (a, b, c) is the direction vector.

Let's choose P1(2, 4, -3) as the **point **on the line. Substituting the values, we get:

x = 2 + t

y = 4 - 5t

z = -3 + 4t

Therefore, the parametric equation of the line **passing **through P1(2, 4, -3) and P(3, -1, 1) is:

x = 2 + t

y = 4 - 5t

z = -3 + 4t

where t is a parameter that varies along the line.

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If 2x2 - 4x + 6 = 1 then A = 2, B = -4, and C = 6 in general form.

### Answers

In **general form**, the equation 2x^2 - 4x + 6 = 1 can be represented as A = 2, B = -4, and C = 5.

To determine the **values **of A, B, and C in the general form of the quadratic equation 2x^2 - 4x + 6 = 1, we can compare the given equation with the standard form of a **quadratic** equation, which is ax^2 + bx + c = 0.

In the given equation, we have:

2x^2 - 4x + 6 = 1

To put it in the general form, we need to move all the **terms **to the left side of the equation, so the right side is equal to 0:

2x^2 - 4x + 6 - 1 = 0

2x^2 - 4x + 5 = 0

Now we can identify the **coefficients **A, B, and C in the general form:

A = 2 (coefficient of x^2)

B = -4 (coefficient of x)

C = 5 (constant term)

The values of A, B, and C in the general form of the **equation **2x^2 - 4x + 6 = 1 are A = 2, B = -4, and C = 5.

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A zoo has categorized its visitors into three categories: member, school, and general. The member category refers to visitors who pay an annual fee to support the zoo. Members

receive certain benefits such as discounts on merchandise and trips planned by the zoo. The school category includes faculty and students from day care and elementary and

secondary schools; these visitors generally receive a discounted rate. The general category includes all other visitors. The zoo has been concerned about a recent drop in

attendance. To help better understand attendance and membership, a zoo staff member has collected the following data.

Attendance

School

Visitor

Category 2011

General 154,213 159,204 163,933 169,606

Member 114,023 103,295

96,937 79,717

82,385 79,376

81,470

80,790

Total 350,621 341,875 342,340

330,113

(0) Construct a bar chart of total attendance over time. Comment on any trend in the data.

(b) Construct a side-by-side bar chart showing attendance by visitor category with year as the variable on the horizontal axis.

(c) Comment on what is happening to zoo attendance based on the charts from parts (a) and (b).

2012

2013

2014

### Answers

(a) Based on the total attendance data provided, a bar chart can be constructed to visualize the attendance trends over time. The horizontal axis represents the years (2011, 2012, 2013, 2014), and the vertical **axis **represents the total attendance. The bars will correspond to the attendance numbers for each year.

(b) A side-by-side bar chart can be created to compare the attendance by visitor category with the year as the variable on the horizontal axis. The visitor categories (school, general, member) will be represented by different colors or patterns, and the bars will show the attendance for each category in each year.

(c) By analyzing the charts from parts (a) and (b), we can observe the trends in zoo attendance.

(a) The bar **chart** of total attendance over time shows the following trend: In 2011, the attendance was 350,621. It decreased slightly in 2012 to 341,875, remained relatively stable in 2013 at 342,340, and then decreased again in 2014 to 330,113.

(b) The side-by-side bar chart showing attendance by visitor category with year as the variable on the horizontal axis provides a visual comparison of attendance for each category over the years.

The chart reveals that the general category consistently had the highest attendance throughout the years, followed by the school category. The member category consistently had the lowest attendance.

(c) Based on the charts from parts (a) and (b), it can be observed that there has been a general decline in zoo attendance over the years, with a notable decrease from 2011 to 2014.

The drop in **attendance** could indicate a potential issue that the zoo needs to address in order to attract more visitors.

Additionally, the consistently lower attendance in the member category suggests that the zoo may need to reconsider its membership benefits and strategies to retain and attract more members.

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What is the area of

the segment? Express

the answer in terms

of pi.

### Answers

The **area** of the **segment** is 9( π-2) units²

What is area of segment?

The **area** of a figure is the number of unit squares that cover the surface of a closed figure.

A **segment** is the **area** occupied by a chord and an arc. A segment can be a major segment or minor segment.

**Area** of **segment** = area of sector - area of triangle

**area** of sector = 90/360 × πr²

= 1/4 × π × 36

= 9π

**area** of triangle = 1/2bh

= 1/2 × 6²

= 18

**area** of **segment** = 9π -18

= 9( π -2) units²

therefore the **area** of the **segment** is 9(π-2) units²

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50 Points! Multiple choice geometry question. Photo attached. Thank you!

### Answers

**Answer:**

B

**Step-by-step explanation:**

SAS Similarity theorem: If two sides in one triangle are proportional to two sides in another triangle and the included angle in both are congruent, then the two triangles are similar.

Side 28 is congruent to side 11.2, whereas side 20 is congruent to side 8 and both angles are congruent. Therefore both triangles are similar.

Which expressions represent the statement divided the difference of 27 and 3 by there difference of 16 and 4

### Answers

The **expression** that represents the statement "Divide the difference of 27 and 3 by the difference of 16 and 4" is (27 - 3) / (16 - 4), which simplifies to 2.

The statement "Divide the **difference **of 27 and 3 by the difference of 16 and 4" can be represented using **algebraic **expressions.

To find the difference between two numbers, we subtract one from the other. So, the difference of 27 and 3 is 27 - 3, which can be expressed as (27 - 3). Similarly, the difference of 16 and 4 is 16 - 4, which can be expressed as (16 - 4).

Now, we need to divide the difference of 27 and 3 by the difference of 16 and 4. We can use the division **operator **(/) to represent the division operation.

Therefore, the expression that represents the given statement is:

(27 - 3) / (16 - 4)

Simplifying this expression further, we have:

24 / 12

The difference of 27 and 3 is 24, and the difference of 16 and 4 is 12. So, the expression simplifies to:

2

Hence, the expression (27 - 3) / (16 - 4) is equivalent to 2.

In summary, the expression that represents the **statement **"Divide the difference of 27 and 3 by the difference of 16 and 4" is (27 - 3) / (16 - 4), which simplifies to 2.

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Two sets that contain exactly the same elements are called ___ sets.

### Answers

Two **sets** that contain exactly the same elements are called "equal sets" or "identical sets."

In set theory, the concept of equality between sets is defined by the **axiom** of extensionality, which states that two sets are equal if and only if they have the same elements.

To illustrate this concept, let's consider two sets: Set A and Set B. If every element of Set A is also an element of Set B, and vice versa, then we say that Set A and Set B are equal sets. In other words, the sets have the exact same **elements**, regardless of their order or repetition.

For example, if Set A = {1, 2, 3} and Set B = {3, 2, 1}, we can observe that both sets contain the same elements, even though the order of the elements is different. Therefore, Set A is equal to Set B.

In summary, equal sets refer to two sets that possess exactly the same elements, without considering the order or repetition of the elements. The concept of equality is fundamental in set **theory** and forms the basis for various operations and theorems in mathematics.

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A random number from 1 to 5 is selected 50 times. The number 1 is selected 13 times, 2 is selected 8 times, 3 is selected 14 times, 4 is selected 6 times, and 5 is selected 9 times. What is the relative frequency of selecting a 2? Express your answer as a percent.

### Answers

**Answer:**

Relative frequency of selecting a 2 = 8/50 = 0.16 = 16%

**Step-by-step explanation:**

You are selecting a random number between 1 and 5, and you perform this task 50 times.

Out of these 50 times, the outcome "2" appears 8 times.

Therefore the relative frequency of selecting the number 2 is:

f(2) = 8/50 = 0.16 which is 16%

I need help understanding how to format this:

f(x)-4

f(x)=2x+1

an example of another question for it (if you're not sure what im asking), another question was to answer this:

f(4)

f(x)=2x+1

f(4)=2(4)+1

this one is easy to get, but i'm not too sure how to put it on the first one...

### Answers

**Answer:**

**Step-by-step explanation:**

To format and solve the equation "f(x) - 4" with the given function "f(x) = 2x + 1," we substitute the function into the equation and solve for x. Here's how it can be done:

f(x) - 4 = 2x + 1 - 4

Simplifying further:

f(x) - 4 = 2x - 3

To answer the question "f(4)" using the function f(x) = 2x + 1, we substitute x = 4 into the function:

f(4) = 2(4) + 1

Simplifying further:

f(4) = 8 + 1

f(4) = 9

Therefore, the value of f(4) is 9 when using the function f(x) = 2x + 1.

Which of the following could be the ratio between the lengths of the two legs

of a 30-60-90 triangle?

Check all that apply.

A. √2:2

B. √√3:√√3

C. √5:3

D. 1 √3

□ E. 1: √2

O F. 2:3

SUBMIT

### Answers

**Answer: E**

**Step-by-step explanation:**

if 3+5 equals 8 then what does 5+3 equal?

### Answers

**Answer:**

8

**Step-by-step explanation:**

Given that √x + √y=138 and x-y=1656, find x.

### Answers

The **value **of x is 5625 when √x + √y=138 and x-y=1656.

To solve for x using the given **equations**, we will use the method of substitution. Let's go through the steps:

Start with the equation √x + √y = 138.

We want to **express **y in terms of x, so solve for y in terms of x by isolating √y:

√y = 138 - √x

Square both sides of the equation to **eliminate **the square root:

(√y)² = (138 - √x)²

y = (138 - √x)²

Now, substitute the expression for y in the second equation x - y = 1656:

x - (138 - √x)² = 1656

Expand the squared term:

x - (138² - 2 * 138 * √x + (√x)²) = 1656

Simplify the equation further:

x - (19044 - 276 * √x + x) = 1656

Combine like terms and rearrange the equation:

-19044 + 276 * √x = 1656

Move the constant term to the other side:

276 * √x = 20700

Divide both sides of the equation by 276 to solve for √x:

√x = 20700 / 276

√x = 75

Square both sides to solve for x:

x = (√x)²

x = 75²

x = 5625

Therefore, the value of x is 5625.

By substituting this value of x back into the original equations, we can verify that √x + √y = 138 and x - y = 1656 hold true.

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The meaningful domain of the linear model are all the possible values the x variable can take

on that make sense. The range is all the possible values for the linear model (the y values).

The top of the mountain is at 8920 feet and the base of the mountain is at 3300 feet

Identify

Domain

Range

### Answers

Domain: The **domain **is the range of valid heights for the mountain, which is from 3300 feet to 8920 feet.

Range: The range is the set of all possible heights of the linear model, which in this case is also from 3300 feet to 8920 feet.

Domain: The domain of the** linear model** in this context would represent the possible values for the x variable, which is associated with the height of the mountain.

In this case, the meaningful domain would be the range of valid heights that the mountain can have.

Since the top of the mountain is at 8920 feet and the base is at 3300 feet, the meaningful domain would be the **range **of heights between 3300 feet and 8920 feet.

Therefore, the domain in this scenario would be [3300, 8920].

Range: The range of the linear model in this context would represent the possible values for the y variable, which is associated with the height of the mountain.

The range would be the set of all **possible heights** that the linear model can produce.

In this case, since the top of the mountain is at 8920 feet and the base is at 3300 feet, the range would encompass all the valid heights within this range.

Therefore, the range in this scenario would be [3300, 8920].

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In this case the linear equations are given:

A company offers two data plans for cell phones. The plan A the linear function for the

charge is given by

y=10x

where x represents the total number of megabytes. The Plan B charge is calculated using

the linear function

y = 4x + 75.

How many megabytes would a customer need to use for Plan be to be a better deal?

1) more than 12.5 megabytes

2) less than 18.75 megabytes

3)

Plan b is always a better deal because the charge per megabyte is less than in

plan A

4) More than 10 megabytes.

5) More than 20 megabytes

### Answers

The customer would need to use more than 12.5 **megabytes **for Plan B to be a better deal. (Option 1) more than 12.5 megabytes).

To determine when Plan B would be a better deal than Plan A, we need to compare the **charges **for both plans based on the number of megabytes used.

Plan A is represented by the **linear function** y = 10x, where x represents the total number of megabytes used, and y represents the charge for the plan.

Plan B is represented by the linear function y = 4x + 75, where x represents the total number of megabytes used, and y represents the charge for the plan.

To find the point at which Plan B becomes a better deal, we need to find the x-value where the charge for Plan B is less than the charge for Plan A.

In other words, we need to find the x-value that satisfies the inequality:

4x + 75 < 10x

To solve this **inequality**, we subtract 4x from both sides:

75 < 6x

Then, we divide both sides by 6:

12.5 < x

Therefore, the customer would need to use more than 12.5 megabytes for Plan B to be a better deal.

This means that option 1) "more than 12.5 megabytes" is the correct answer.

For any value of x greater than 12.5, the charge for Plan B will be less than the charge for Plan A, making it a better deal for the customer.

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John Doe produces two kinds of men’s shirts: polo and t-shirts. Polo shirts require 2 hours in the pattern and cutting section and 1 hour in the sewing section. T-shirts require 1 hours in the pattern and cutting section and 2 hours in the sewing section. The pattern and cutting section has 84 hours available weekly. The sewing section has 106 hours available weekly. Past sales indicate that at most 36 polo shirts can be sold. The profit on each polo shirt is $30 and on each t-shirt is $22. How many of each kind should the company produce in order to maximize its profit?

a) Define your variables (2):

b) Constraints (5):

c) Objective function (1):

d) Graph (label the axes) and Work

### Answers

The **evaluation** of the constraints** **with regards to the **production** of the polo and t-shirts, and to **maximize** the profit, using **linear programming **indicates that we get;

a. x = The number of polo shirts produced, y = The number of t-shirts produced

b. The inequalities representing the constraints are;

2·x + y ≤ 84

x + 2·y ≤ 106

x ≤ 36

x ≥ 0, y ≥ 0

c. P = 30·x + 22·y

d. Please find attached the graph of the feasible region

To maximize profit, the company should produce;

21 polo shirts and 43 t-shirts

What is linear programming?

**Linear programming** is a method used for **optimizing** (maximizing or minimizing a value) operations with some specified constraints.

a. The details indicates that the question is related to** linear programming**. Let *x* **represent** the number of polo shirt produced, and let *y* represent the number of t-shirts produced.

x = The number of polo shirt produced

y = The number of t-shirts produced

b) The constraints are;

Pattern and cutting section; 2·x + y ≤ 84

Sewing section; x + 2·y ≤ 106

Sales **constraints**; *x* ≤ 36

The values of *x* and *y* are non negative, numbers, therefore;

x ≥ 0, y ≥ 0

c) The **objective** of the company is to **maximize** **profit**, *P*, therefore, the objective function is; *P* = 30·x + 22·y

d) The **graph** can be plotted from the **constraint inequalities**, by making *y* the subject in the inequalities that includes both *x* and *y* as follows;

2·x + y ≤ 84, therefore; y ≤ 84 - 2·x

x + 2·y ≤ 106, therefore; y ≤ 53 - x/2

*x* ≤ 36

Please find attached the **graph** of the **inequalities**, showing the** feasible region** which is the **polygon** with boundaries which are the lines representing the constraints.

The objective function evaluated at the vertices of the feasible region indicates that we get;

[tex]\begin{tabular}{ | l | l | c | }\cline{1-3}(x, y)& 30\cdot x + 22\cdot y & P(\$) \\ \cline{1-3}(0, 53 & 30\times 0 + 22\times 53 & 1166 \\\cline{1-3}(21, 42.5 & 30\times 21 + 22\times 42.5 & 1565 \\\cline{1-3}(32, 12) & 30\times 36 + 22\times 12 & 1344 \\\cline{1-3}(36, 0) & 30\times 36 + 22\times 0 & 1080 \\\cline{1-3}(0, 0) & 30\times 0 + 22\times 0& 0 \\\cline{1-3}\end{tabular}[/tex]

The feasible region and the objective function indicates that the values of *x* and *y* that maximizes the profit is; (x, y) = (21, 42.5)

Therefore, to maximize profit, the number of polo and t-shirts the company should produce are 21, and 42.5 ≈ 43 respectively.

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What is the distance from A to B?

10

108642

A(-8, -3)

-2

-6

-8

-10

B6, 6)

2 4 6 8 10

A 21 units

B. 15 units

C. 225 units

D. 3 units

### Answers

The **distance **from point A to point B is approximately 16.64 units. None of the given options (A, B, C, D) match this value exactly, so there seems to be an error in the options provided.

To find the distance from **point **A to point B, we can use the distance formula in Euclidean geometry. The distance formula **between **two points (x1, y1) and (x2, y2) is given by:

distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

In this case, point A is (-8, -3) and point B is (6, 6). Plugging these values into the distance **formula**, we have:

distance = sqrt((6 - (-8))^2 + (6 - (-3))^2)

= sqrt((6 + 8)^2 + (6 + 3)^2)

= sqrt(14^2 + 9^2)

= sqrt(196 + 81)

= sqrt(277)

≈ 16.64

Thus, the distance between points A and B is roughly 16.64 units. Since none of the **available **options (A, B, C, or D) exactly match this value, there appears to be a problem with the options.

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5 In a Survery of 130 people 80 claimed to be CDO partisans and 60 claimed to be Anc partisan. If 30 of them are both ANC and CDO how many people are none of these two parties

### Answers

**Answer: there are 20 people who claimed to be neither CDO partisans nor ANC partisans.**

**Step-by-step explanation:**

To determine the number of people who are none of these two parties, we need to subtract the total number of people who claimed to be CDO partisans, ANC partisans, and those who claimed to be both from the total number of people surveyed.

Total surveyed people = 130

Number claiming to be CDO partisans = 80

Number claiming to be ANC partisans = 60

Number claiming to be both ANC and CDO = 30

To find the number of people who are none of these two parties, we can calculate it as follows:

None of these two parties = Total surveyed people - (CDO partisans + ANC partisans - Both ANC and CDO)

None of these two parties = 130 - (80 + 60 - 30)

None of these two parties = 130 - 110

None of these two parties = 20

the value of tan80°×tan10°+sin70+sin 20

### Answers

The **value** of tan(80°) × tan(10°) + sin(70°) + sin(20°) = 2.28171276

What is Trigonometry?

**Trigonometry **is the** branch of mathematics **concerned with specific functions of angles and their application to calculations. There are six **functions **of an angle commonly used in** trigonometry**. Their names and abbreviations are sine (**sin**), cosine (**cos**), tangent (**tan**), cotangent (**cot**), secant (**sec**), and cosecant (**csc**). These six **trigonometric functions **in relation to a right triangle are displayed in the figure.

Given:

tan(80°) × tan(10°) + sin(70°) + sin(20°)

Using **Trigonometric identity**

sin (A · B) = sin A cos B - cos A sin B

So, tan(80°) × tan(10°) + sin(70°) + sin(20°)

[tex]\rightarrow \tan (80 \times 10)=1[/tex]

[tex]\rightarrow\sin(70^\circ+20^\circ)=1.28171276[/tex]

[tex]\rightarrow 1.28171276+1=\bold{2.28171276}[/tex]

Hence, tan(80°) × tan(10°) + sin(70°) + sin(20°) = 2.28171276

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