Chapter 4, Problem 1STP

To determine

To find: The correct option of the price that will maximise the income of the store.

Answer to Problem 1STP

The correct answer is C.

Explanation of Solution

Given:

The given information is:

A rental store charges $\$30$ per day to rent a bicycle.

The store rents about $200$ bikes per month.

  $\text{Let the number of increases in the rental prices of a bike}=x$ .

So the new price is $30+2x$ .

The new number of clients is $200-5x$ .

The function which determines the income is given as:

  $\begin{array}{c}f\left(x\right)=\left(30+2x\right)\left(200-5x\right)\\ f\left(x\right)=6000+400x-150x-10x^2\\ f\left(x\right)=-10x^2+250x+6000\end{array}$

The graph of the above function is given as:

  

From the above graph it is clear that the above function attains its maximum value at $x=12.5$ .

The new price to maximise the income is given as:

  $30+2\left(12.5\right)=55$ .

So the price that will maximise the income of the store is $\$55$ .

Therefore the correct answer is C.