Read the scenario. Then, use the simplified $($continuous$)$ CLV formula to answer the following questions.

Scenario:

A local gym has $500$ current members.

The gym has an annual membership fee of $$800.($Note: This is annualized.$)$

Variable costs per year per member is $$100.$

The gym reports a member retention rate of $90\%.$

Members complain that the locker rooms are ugly and outdated.

Assume a current discount rate of $10\%.$

What is the current lifetime value per customer?

Consider how the gym should best invest $$10,000.$ They can choose either of two options:

Option A: Upgrade the locker room

A locker room facelift is estimated to increase the retention rate from $90\%$ to $92\%.$

Stay at $500$ members; increase CLV by increasing retention rate by $2\%.$

Option B: Advertise for new members

An intense advertising effort is estimated to attract $50$ new members $($i$.$e$.,$ increase membership by $10\%).$

Gain $50$ new members; keep CLV the same.

What is the net profit that will come from each option? Which would you select? Why?

Explain the results that you calculated. Did the results come out as you would expect? Why or why not?