Question

to join a rock climbing gym you must pay an initial fee of $100 and a monthly fee of $59a. how many months must you have the membership for the average cost per month to fall to $69?b. what happens to the average cost as the number of months that you are a member increases?

Answer 1

We have a gym fee that is divided into an initial fee of $100 and a monthly fee of $59.

Let n be the number of months of membership. Then, we can express the total cost for the membership as:

`C(n)=100+59n`

a) The average cost per month can be calculated as the total cost C(n) divided by the number of months n.

We can express it as:

`(C(n))/(n)=(100+59n)/(n)=(100)/(n)+59`

Then, we can find the number of months for which C(n)/n = 69 as:

`\begin{gathered} (C(n))/(n)=69 \\ (100)/(n)+59=69 \\ (100)/(n)=69-59 \\ (100)/(n)=10 \\ 100=10\cdot n \\ n=(100)/(10) \\ n=10 \end{gathered}`

After 10 months the average cost will be $69.

b) We can calculate this as the limit of C(n)/n when n increases infinitely. This will give us the average value for the long run.

We can calculate it as:

`\begin{gathered} \lim _(n\to\infty)(C(n))/(n) \\ \lim _(n\to\infty)((100)/(n)+59)=0+59=59 \end{gathered}`

When n increases, the initial fee "dilutes" its influence in the average value while the monthly fee stays constant for the average value.

Then, as the number of months increases, the average cost approaches the monthly fee.

Answer:

a) 10 months

b) the average cost approaches the monthly fee ($59)