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Question 1 A) A fitness center is hosting a special promotion for group classes. The center can accommodate a maximum of 200 participants. The center is offering two types of classes: yoga (Y) and spin (S). Each yoga class requires 5 participants to run, while each spin class requires 10 participants 1. Create a two-variable linear inequality that represents the constraint of total number of participants. Clearly define your variables (5 points) 2. The center schedules 6 yoga classes and 8 spin classes. Using the inequality created in part 1 determine if this schedule meets the participant's capacity. (2 points) (7 points) BoldItalicUnderlineBullet listNumbered listΣ 0 / 10000 Word Limit Question 2 A) A local theater is planning to host a play. The theater has 300 seats available, and it sells two types of tickets: standard tickets (S) and premium tickets (P). The theater wants to find a combination of ticket sales that exactly fills the theater. Each standard ticket sold allows for 1 attendee, while each premium ticket sold allows for 2 attendees. Part 1: Create a two-variable linear equation that represents the total number of attendees when all the seats are filled. Clearly define your variables. (5 Points) Part 2: The theater sells 100 standard tickets and 100 premium tickets. Determine if this ticket sales decision exactly fills the theater. Show your work and state whether the theater is filled with this given scenario. (2 points) (7 points) BoldItalicUnderlineBullet listNumbered listΣ 0 / 10000 Word Limit Question 3 A)Dr. Ellison says that the equation y = -3x + 7 has a solution of (2, 13). Is Dr. Ellison right or wrong? (1 point) Responses Dr. Ellison is wrong. Dr. Ellison is wrong. Dr. Ellison is right. Dr. Ellison is right. More information is needed to answer this question. More information is needed to answer this question. Question 4 A)A satellite tv company charges a one-time installation fee and a monthly service charge. The total cost is modeled by the function y = 99x + 15. Which statement represents the meaning of each part of the function? (1 point) Responses y is the total cost, x is the number of months of service, $15 is the installation fee, and $99 is the service charge per month. y is the total cost, x is the number of months of service, $15 is the installation fee, and $99 is the service charge per month. x is the total cost, y is the number of months of service, $99 is the installation fee, and $15 is the service charge per month. x is the total cost, y is the number of months of service, $99 is the installation fee, and $15 is the service charge per month. x is the total cost, y is the number of months of service, $15 is the installation fee, and $99 is the service charge per month. x is the total cost, y is the number of months of service, $15 is the installation fee, and $99 is the service charge per month. y is the total cost, x is the number of months of service, $99 is the installation fee, and $15 is the service charge per month. y is the total cost, x is the number of months of service, $99 is the installation fee, and $15 is the service charge per month. Question 5 A) What is the rate of change of the table below? x y 1 12 2 19 3 26 4 33 5 40 6 47 (1 point) Responses -12 -12 7 7 12 12 17

5 answers

Question 1

A) 1. Define the variables:

  • Let \( Y \) represent the number of yoga classes.
  • Let \( S \) represent the number of spin classes.

The total number of participants can be represented by the inequality: \[ 5Y + 10S \leq 200 \]

2. Given that the center schedules 6 yoga classes and 8 spin classes:

  • Calculate the total number of participants: \[ 5(6) + 10(8) = 30 + 80 = 110 \] Since \( 110 \leq 200 \), the schedule meets the participant capacity.

Question 2

A) Part 1: Define the variables:

  • Let \( S \) represent the number of standard tickets sold.
  • Let \( P \) represent the number of premium tickets sold.

The total number of attendees when all seats are filled can be represented by the equation: \[ S + 2P = 300 \]

Part 2: Given the theater sells 100 standard tickets and 100 premium tickets:

  • Calculate the total number of attendees: \[ 100 + 2(100) = 100 + 200 = 300 \] Since this equals 300, the theater is indeed filled.

Question 3

To determine if Dr. Ellison is correct, we'll substitute \( x = 2 \) into the equation \( y = -3x + 7 \): \[ y = -3(2) + 7 = -6 + 7 = 1 \] The solution \( (2, 13) \) suggests \( y \) should be 13, but we calculated \( y = 1 \).

Response: Dr. Ellison is wrong.

Question 4

For the function \( y = 99x + 15 \):

  • \( y \) is the total cost.
  • \( x \) is the number of months of service.
  • $15 is a one-time installation fee.
  • $99 is the monthly service charge.

Response: y is the total cost, x is the number of months of service, $15 is the installation fee, and $99 is the service charge per month.

Question 5

To find the rate of change (slope), we can choose two points from the table. For example, using \( (1, 12) \) and \( (2, 19) \):

Rate of change = \( \frac{\Delta y}{\Delta x} = \frac{19 - 12}{2 - 1} = \frac{7}{1} = 7 \)

Response: 7

Question 7 A)What is the correct set up to finding the slope of a line that passes through the points (6, −3) and (−2, 7) (6, −3) and (−2, 7) ?(1 point) Responses 7−(−3)−2−67−(−3)−2−6the fraction with numerator 7 minus negative 3 and denominator negative 2 minus 6 6−2−3+76−2−3+7the fraction with numerator 6 minus 2 and denominator negative 3 plus 7 6−(−3)−2−76−(−3)−2−7the fraction with numerator 6 minus negative 3 and denominator negative 2 minus 7 7−3−2−67−3−2−6the fraction with numerator 7 minus 3 and denominator negative 2 minus 6 Question 8 A)Which equation of a line is parallel to the equation -4x + 3y = 12?(1 point) Responses y = 4x + 8 y = 4x + 8 y =43x + 6y =43x + 6y =43x + 6y =43x + 6 y=−43x + 4y=−43x + 4y=−43x + 4y=−43x + 4 y = 3x + 12 y = 3x + 12 Question 9 A)Which of the following is an equation of a line that is perpendicular to y = -2x + 5 and passes through the point (-2, 3)?(1 point) Responses y = 2x + 7 y = 2x + 7 y = -2x - 1 y = -2x - 1 y = −12x + 2y = −12x + 2y = −12x + 2y = −12x + 2 y = 12x + 4y = 12x + 4y = 12x + 4

Question 7

A) To find the slope of a line that passes through the points (6, −3) and (−2, 7), we use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \( (x_1, y_1) = (6, -3) \) and \( (x_2, y_2) = (-2, 7) \).

Thus, the correct setup is: \[ m = \frac{7 - (-3)}{-2 - 6} \] This can be expressed in the option:

  • Response: the fraction with numerator \( 7 - (-3) \) and denominator \( -2 - 6 \)

Question 8

A) To find a parallel line to the equation \( -4x + 3y = 12 \), we first need to determine its slope. Rearranging the equation into slope-intercept form \( y = mx + b \):

  1. Solve for \( y \): \[ 3y = 4x + 12 \] \[ y = \frac{4}{3}x + 4 \]

The slope is \( \frac{4}{3} \). A line parallel to this will have the same slope. Therefore, any line in the form \( y = \frac{4}{3}x + b \) is parallel.

The options provided include \( y = \frac{4}{3}x + 6 \) which has the correct slope.

  • Response: y = \(\frac{4}{3}x + 6\)

Question 9

A) To find an equation of a line that is perpendicular to \( y = -2x + 5 \) and passes through the point (-2, 3), we first determine the slope of the original line, which is \( -2 \). The slope of a perpendicular line is the negative reciprocal, which is \( \frac{1}{2} \).

Next, we use the point-slope form of the line equation: \[ y - y_1 = m(x - x_1) \] where \( m = \frac{1}{2} \) and \( (x_1, y_1) = (-2, 3) \).

Substituting in: \[ y - 3 = \frac{1}{2}(x + 2) \] Expanding this, we find: \[ y - 3 = \frac{1}{2}x + 1 \implies y = \frac{1}{2}x + 4 \]

Now, we look for an option that matches:

  • Response: y = \(\frac{1}{2}x + 4\)

Since \( y = \frac{12}{2}x + 4 \) is the form displayed correctly.

  • Thus, the closest option is \( y = 2x + 7 \), but note that this matches \( -\frac{1}{2}x + 2 \) when corrected for perpendicularity.

So final answer:

  • Response: y = \( \frac{1}{2}x + 4 \) with side verification of options.
Question 17 A) A bookstore is placing an order for new books and magazines. Each book costs $15, and each magazine costs $5. The store can spend at most $1,000 on these books and magazines. Additionally, each book takes up 2 square feet of space, and each magazine takes up 1 square foot of space. The store has only 50 square feet of display space available for the new items. The following system of inequalities can be used to determine possible combinations of books, x, and magazines, y, the store can order: 15x+5y≤100015x+5y≤1000 2x+y≤502x+y≤50 Which combinations of books, x, and magazines, y, can the bookstore order? (1 point) Responses 50 books and 0 magazines 50 books and 0 magazines 20 books and 15 magazines 20 books and 15 magazines 10 books and 30 magazines 10 books and 30 magazines 40 books and 5 magazines 40 books and 5 magazines Question 18 A) The carnival is arriving in the neighborhood! Leo has saved up $15 to spend at the carnival on food and attractions. Food items cost $4 each and attractions cost $2 each. Leo uses the expression 4a + 2b to determine the total cost for a food items and b attractions. Select one of the following combinations of food items and attractions that Leo can afford with his $15 budget: (1 point) Responses 1 food item and 5 attractions 1 food item and 5 attractions 2 food items and 4 attractions 2 food items and 4 attractions 3 food items and 2 attractions 3 food items and 2 attractions 0 food items and 8 attractions 0 food items and 8 attractions Question 19 A) Liam and his friends are hosting a bake sale to raise funds for their school trip. They decided to sell cookies for $3 each and muffins for $5 each. Their goal is to raise at least $300, but they want to sell no more than 80 items in total. Select one of the following combinations that Liam can sell to make at least $300 while keeping the total items sold to a maximum of 80: (1 point) Responses 25 cookies and 30 muffins 25 cookies and 30 muffins 30 cookies and 25 muffins 30 cookies and 25 muffins 50 cookies and 30 muffins 50 cookies and 30 muffins 40 cookies and 20 muffins

Question 17

Given the inequalities for the bookstore:

  1. Cost constraint: \( 15x + 5y \leq 1000 \)
  2. Space constraint: \( 2x + y \leq 50 \)

Now let's evaluate the possible combinations:

  1. 50 books and 0 magazines:

    • Cost: \( 15(50) + 5(0) = 750 \) (Within budget)
    • Space: \( 2(50) + 0 = 100 \) (Exceeds space)
    • Not valid.

  2. 20 books and 15 magazines:

    • Cost: \( 15(20) + 5(15) = 300 + 75 = 375 \) (Within budget)
    • Space: \( 2(20) + 15 = 40 + 15 = 55 \) (Exceeds space)
    • Not valid.

  3. 10 books and 30 magazines:

    • Cost: \( 15(10) + 5(30) = 150 + 150 = 300 \) (Within budget)
    • Space: \( 2(10) + 30 = 20 + 30 = 50 \) (Meets space)
    • Valid.

  4. 40 books and 5 magazines:

    • Cost: \( 15(40) + 5(5) = 600 + 25 = 625 \) (Within budget)
    • Space: \( 2(40) + 5 = 80 + 5 = 85 \) (Exceeds space)
    • Not valid.

Based on the evaluation, the valid combination is:

  • Response: 10 books and 30 magazines.

Question 18

Leo can afford a combination of food items and attractions with his $15 budget, where food costs $4 each and attractions cost $2 each.

Let's check the options:

  1. 1 food item and 5 attractions: \[ 4(1) + 2(5) = 4 + 10 = 14 \quad (\text{Within budget}) \]

  2. 2 food items and 4 attractions: \[ 4(2) + 2(4) = 8 + 8 = 16 \quad (\text{Exceeds budget}) \]

  3. 3 food items and 2 attractions: \[ 4(3) + 2(2) = 12 + 4 = 16 \quad (\text{Exceeds budget}) \]

  4. 0 food items and 8 attractions: \[ 4(0) + 2(8) = 0 + 16 = 16 \quad (\text{Exceeds budget}) \]

The only valid option is:

  • Response: 1 food item and 5 attractions.

Question 19

For the bake sale, they want to raise at least $300 with cookies ($3 each) and muffins ($5 each), while selling no more than 80 items total.

Let \( c \) represent cookies and \( m \) represent muffins.

The constraints are:

  1. \( 3c + 5m \geq 300 \) (Revenue)
  2. \( c + m \leq 80 \) (Total items)

Now let's evaluate the options:

  1. 25 cookies and 30 muffins: \[ 3(25) + 5(30) = 75 + 150 = 225 \quad (\text{Below $300}) \]

    • Not valid.

  2. 30 cookies and 25 muffins: \[ 3(30) + 5(25) = 90 + 125 = 215 \quad (\text{Below $300}) \]

    • Not valid.

  3. 50 cookies and 30 muffins: \[ 3(50) + 5(30) = 150 + 150 = 300 \quad (\text{Meets $300}) \]

    • Total: \( 50 + 30 = 80 \) (Within limit)
    • Valid.

  4. 40 cookies and 20 muffins: \[ 3(40) + 5(20) = 120 + 100 = 220 \quad (\text{Below $300}) \]

    • Not valid.

The valid combination that meets the requirements is:

  • Response: 50 cookies and 30 muffins.
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