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 0 | 2 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

Explanation:

Step1: Find the equation for Gym A

First, we determine the slope \( m \) of the linear function for Gym A using the formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Taking the first two points \((0, 2)\) and \((4, 7)\), we have \( m=\frac{7 - 2}{4 - 0}=\frac{5}{4} = 1.25 \). The y - intercept \( b \) is 2 (when \( x = 0 \), \( y=2 \)). So the equation for Gym A is \( y = 1.25x+2 \).

Step2: Find the equation for Gym B

For Gym B, the y - intercept \( b \) is 4 (when \( x = 0 \), \( y = 4 \)). The slope \( m \) is calculated as the rate of change. It increases by 3 when \( x \) increases by 4, so \( m=\frac{3}{4}=0.75 \). The equation for Gym B is \( y=0.75x + 4 \).

Step3: Analyze the cost functions

To find when the costs are equal, we set the two equations equal to each other: \( 1.25x+2=0.75x + 4 \). Subtract \( 0.75x \) from both sides: \( 1.25x-0.75x+2=4 \), which simplifies to \( 0.5x+2 = 4 \). Subtract 2 from both sides: \( 0.5x=2 \). Multiply both sides by 2: \( x = 4 \). When \( x = 4 \), for Gym A: \( y=1.25\times4 + 2=5 + 2=7 \). For Gym B: \( y=0.75\times4+4=3 + 4 = 7 \). For \( x<4 \), let's take \( x = 0 \): Gym A cost is 2, Gym B cost is 4, so Gym A is cheaper. For \( x>4 \), let's take \( x = 8 \): Gym A cost is \( 1.25\times8+2=10 + 2 = 12 \), Gym B cost is \( 0.75\times8+4=6 + 4=10 \), so Gym B is cheaper.

(If the question was to find when the costs are equal or which is cheaper at a certain \( x \), this is the analysis. Since the question was not fully specified, but based on the work to show, this is the process of finding the cost functions and comparing them.)

Answer:

The equation for Gym A is \( y = 1.25x + 2 \), for Gym B is \( y=0.75x + 4 \). They cost the same at \( x = 4 \) hours (cost \( \$7 \)), Gym A is cheaper for \( x<4 \), Gym B is cheaper for \( x>4 \).