Question

milo is deciding whether to join gym a or gym b based upon each of their membership plans.
the membership cost at gym a is given in the following table, with x representing the number of hours of
milo spends at workout classes, and with y representing the total cost in dollars.

x	y
4	7
8	12
12	17
16	22

let the membership cost at gym b be a linear function, with x representing the number of hours of milo
spends at workout classes and y representing the total cost in dollars. the function has a value of 4 when
x = 0, and it increases by 3 each time x increases by 4.

show your work here

which gym has a higher base price (no additional workout classes)?

○ gym a ○ gym b

after how many hours of workout classes do the two gym memberships cost the same amount? round your answer to
two decimal places if necessary.

Explanation:

Step1: Find base price of Gym A

The base price is when \( x = 0 \). From Gym A's table, when \( x = 0 \), \( y = 2 \). So Gym A's base price is $2.

Step2: Find base price of Gym B

For Gym B, when \( x = 0 \), \( y = 4 \) (given). So Gym B's base price is $4.

Step3: Compare base prices

Since \( 4 > 2 \), Gym B has a higher base price.

Step4: Find equations for both gyms

Gym A:

The table shows a linear relationship. Let's find the slope \( m_A \). Using two points, e.g., \( (0, 2) \) and \( (4, 7) \).
Slope \( m_A=\frac{7 - 2}{4 - 0}=\frac{5}{4}=1.25 \).
Equation: \( y_A = 1.25x + 2 \) (since \( y \)-intercept \( b_A = 2 \)).

Gym B:

Slope \( m_B \): Increases by 3 when \( x \) increases by 4, so \( m_B=\frac{3}{4}=0.75 \).
Equation: \( y_B = 0.75x + 4 \) (since \( y \)-intercept \( b_B = 4 \)).

Step5: Find when costs are equal

Set \( y_A = y_B \):
\( 1.25x + 2 = 0.75x + 4 \)
Subtract \( 0.75x \) from both sides: \( 0.5x + 2 = 4 \)
Subtract 2: \( 0.5x = 2 \)
Divide by 0.5: \( x = 4 \)

Answer:

1. Higher base price: Gym B
2. Equal cost at \( \boldsymbol{4.00} \) hours.