membership plans\nmax visits per month cost per visit ($) monthly fee ($)\n12 5 15\n16 4 32\n24 3…

membership plans\nmax visits per month cost per visit ($) monthly fee ($)\n12 5 15\n16 4 32\n24 3 42\nkaleigh wants to know how much she will pay for her gym membership. which graph represents the total monthly cost at the gym as a function of the number of visits kaleigh plans to make each month?

membership plans\nmax visits per month cost per visit ($) monthly fee ($)\n12 5 15\n16 4 32\n24 3 42\nkaleigh wants to know how much she will pay for her gym membership. which graph represents the total monthly cost at the gym as a function of the number of visits kaleigh plans to make each month?

Answer

Answer:

We need to find the cost - function for each membership plan and then analyze the graphs.

  1. For the first plan ($12$ max visits per month):
    • The cost function $C_1(x)$ for the number of visits $x$ ($0\leq x\leq12$) is a linear function of the form $C_1(x)=5x + 15$. When $x = 0$, $C_1(0)=15$ (the monthly fee). When $x = 12$, $C_1(12)=5\times12+15=60 + 15=75$.
  2. For the second plan ($16$ max visits per month):
    • The cost function $C_2(x)$ for $12\lt x\leq16$ is $C_2(x)=4x + 32$. When $x = 12$, $C_2(12)=4\times12+32=48 + 32=80$ (note the non - smooth transition from the first plan). When $x = 16$, $C_2(16)=4\times16+32=64 + 32=96$.
  3. For the third plan ($24$ max visits per month):
    • The cost function $C_3(x)$ for $16\lt x\leq24$ is $C_3(x)=3x + 42$. When $x = 16$, $C_3(16)=3\times16+42=48+42 = 90$ (non - smooth transition from the second plan). When $x = 24$, $C_3(24)=3\times24+42=72 + 42=114$.

The graph will have a non - smooth (piece - wise linear) shape with different slopes for each segment corresponding to the different cost - per - visit and monthly fees of each plan. We need to look for a graph that has three linear segments with the correct starting and ending points and slopes as calculated above.

Since the graphs are not labeled, we can't directly pick a specific graph. But the general characteristics of the correct graph are:

  • It starts at the $y$ - intercept equal to the lowest monthly fee ($y = 15$ when $x = 0$).
  • It has a slope of $5$ for $0\leq x\leq12$, then a slope of $4$ for $12\lt x\leq16$, and a slope of $3$ for $16\lt x\leq24$.

Explanation:

Step1: Define cost functions for each plan

For the first plan with max visits $12$, cost per visit $5$ and monthly fee $15$, $C_1(x)=5x + 15,0\leq x\leq12$. For the second plan with max visits $16$, cost per visit $4$ and monthly fee $32$, $C_2(x)=4x + 32,12\lt x\leq16$. For the third plan with max visits $24$, cost per visit $3$ and monthly fee $42$, $C_3(x)=3x + 42,16\lt x\leq24$.

Step2: Calculate key points

For $C_1(x)$: When $x = 0$, $C_1(0)=15$; when $x = 12$, $C_1(12)=75$. For $C_2(x)$: When $x = 12$, $C_2(12)=80$; when $x = 16$, $C_2(16)=96$. For $C_3(x)$: When $x = 16$, $C_3(16)=90$; when $x = 24$, $C_3(24)=114$.

Step3: Analyze graph characteristics

The graph should be piece - wise linear with $y$ - intercept $15$, slopes $5$, $4$ and $3$ for different intervals of $x$ values and non - smooth transitions at $x = 12$ and $x = 16$.