Chapter SH, Problem 4.22EP

a.

To determine

Find the vertex form of the function using given data.

Answer to Problem 4.22EP

The vertex form of the function is $S\left(p\right)=-2{\left(p-20\right)}^2+2050$ .

Explanation of Solution

Given:

The given modelis $S\left(p\right)=-2p^2+80p+1250$ , where $S\left(p\right)$ is amount of sales of the video game company and $p$ is the price per game.

Calculation:

The vertex form of the equation is $y=a{\left(x-h\right)}^2+k$ , where $\left(h,k\right)$ is the vertex of the parabola.

Write the original equation.

  $S\left(p\right)=-2p^2+80p+1250$

Use distributive property.

  $\Rightarrow S\left(p\right)=-2\left(p^2-40p\right)+1250$

Complete the square.

  $\Rightarrow S\left(p\right)=-2\left(p^2-40p+{20}^2\right)+1250+2\cdot {20}^2$

Simplify.

  $\begin{array}{l}\Rightarrow S\left(p\right)=-2{\left(p-20\right)}^2+1250+800\\ \Rightarrow S\left(p\right)=-2{\left(p-20\right)}^2+2050\end{array}$

Hence the vertex form of the function is $S\left(p\right)=-2{\left(p-20\right)}^2+2050$ .

b.

To determine

Find the maximum amount of daily sales the company make.

Answer to Problem 4.22EP

The maximum amount of daily sales the company make is $\$2050$ .

Explanation of Solution

Given:

The given model is $S\left(p\right)=-2p^2+80p+1250$ , where $S\left(p\right)$ is amount of sales of the video game company and $p$ is the price per game.

Calculation:

The vertex form of the equation is $y=a{\left(x-h\right)}^2+k$ , where $\left(h,k\right)$ is the vertex of the parabola.

Write the original equation.

  $S\left(p\right)=-2p^2+80p+1250$

Use distributive property.

  $\Rightarrow S\left(p\right)=-2\left(p^2-40p\right)+1250$

Complete the square.

  $\Rightarrow S\left(p\right)=-2\left(p^2-40p+{20}^2\right)+1250+2\cdot {20}^2$

Simplify.

  $\begin{array}{l}\Rightarrow S\left(p\right)=-2{\left(p-20\right)}^2+1250+800\\ \Rightarrow S\left(p\right)=-2{\left(p-20\right)}^2+2050\end{array}$

Draw a graph for the given model $S\left(p\right)=-2{\left(p-20\right)}^2+2050$ .

Vertex $=\left(20,2050\right)$

Draw a table for the values of $S\left(p\right)$ with respect to $p$ .

$p$	$S\left(p\right)=-2p^2+80p+1250$	$\left(t,h\right)$
$\begin{array}{l}-12.016\\ 0\\ 20\\ 40\\ 52.016\end{array}$	$\begin{array}{l}0\\ 1250\\ 2050\\ 1250\\ 0\end{array}$	$\begin{array}{l}\left(-12.016,0\right)\\ \left(0,1250\right)\\ \left(20,2050\right)\\ \left(40,1250\right)\\ \left(52.016,0\right)\end{array}$

The graph of the function is given below.

  

From the graph.

For the maximum amount of daily sales $p=20$ .

At $p=20$ , the value of $S\left(p\right)=2050$ .

Hence the maximum amount of daily sales the company make is $\$2050$ .