a high school student is considering whether or not to purchase a pool pass for the summer. without a pass…

a high school student is considering whether or not to purchase a pool pass for the summer. without a pass, the student would pay $12 per visit. with the purchase of a $200 pass, the student would pay $4 per visit. compare the two options to determine which option costs the least for x number of visits. explain or show calculations to support the answer. include equations that model each option in the response. enter the equations, explanation or calculations, and answer in the box.

a high school student is considering whether or not to purchase a pool pass for the summer. without a pass, the student would pay $12 per visit. with the purchase of a $200 pass, the student would pay $4 per visit. compare the two options to determine which option costs the least for x number of visits. explain or show calculations to support the answer. include equations that model each option in the response. enter the equations, explanation or calculations, and answer in the box.

Answer

Explanation:

Step1: Define cost - equations

Let $x$ be the number of visits. The cost without a pass, $C_1$, is given by the equation $C_1 = 12x$ (since it's $12$ dollars per visit). The cost with a pass, $C_2$, is given by the equation $C_2=200 + 4x$ (a fixed - cost of $200$ plus $4$ dollars per visit).

Step2: Set up an inequality to find when $C_1>C_2$

We want to find when $12x>200 + 4x$. Subtract $4x$ from both sides: $12x-4x>200 + 4x-4x$ $8x>200$. Divide both sides by $8$: $x>\frac{200}{8}=25$.

Step3: Analyze the results

If $x < 25$, then $C_1=12x<200 + 4x = C_2$. For example, if $x = 24$, $C_1=12\times24 = 288$ and $C_2=200+4\times24=200 + 96=296$. If $x = 25$, then $C_1=12\times25=300$ and $C_2=200 + 4\times25=200+100 = 300$. If $x>25$, then $C_1=12x>200 + 4x = C_2$. For example, if $x = 26$, $C_1=12\times26 = 312$ and $C_2=200+4\times26=200 + 104=304$.

Answer:

The equation for the cost without a pass is $C_1 = 12x$. The equation for the cost with a pass is $C_2=200 + 4x$. If the number of visits $x<25$, not purchasing the pass is cheaper. If $x = 25$, both options cost the same. If $x>25$, purchasing the pass is cheaper.