Chapter 8.4, Problem 48STP

To determine

To calculate the number of cartridges sold at $20 per cartridge

Answer to Problem 48STP

The number of cartridge sold at $20 per cartridge is $60,000$ . The correct answer is option H.

Explanation of Solution

Given information:

The function of the projected sales of a game cartridge is $S\left(p\right)=\frac{3000}{2p+a}$ . Here $S\left(p\right)$ is the number of cartridges sold, in thousands and $p$ is the price per cartridge, in dollars and $a$ is constant.

The $100,000$ cartridges are sold at $10 per cartridge.

The given options are,

F. $20,000$

G. $50,000$

H. $60,000$

J. $150,000$

At first solve the given function and find the constant $a$ . It is given that $100,000$ video games are sold at $10 each. Since $S\left(p\right)$ is in thousands so the function can be solved as,

  $\begin{array}{l}S\left(p\right)=\frac{3000}{2p+a}\\ 100=\frac{3000}{2\left(10\right)+a}\\ 20+a=\frac{3000}{100}\\ 20+a=30\\ a=30-20\\ a=10\end{array}$

Now, the constant $a$ is known so the number of cartridges sold at $20 per cartridges can be calculated as,

  $\begin{array}{l}S\left(p\right)=\frac{3000}{2\left(20\right)+10}\\ S\left(p\right)=\frac{3000}{40+10}\\ S\left(p\right)=\frac{3000}{50}\\ S\left(p\right)=60\end{array}$

Since the $S\left(p\right)$ is in thousands therefore the answer is $60,000$