Q. 104 - Precalculus Enhanced with Graphing Utilities

Question

Profit Function Suppose that the revenue $R$, in dollars, from selling x clocks is $R\left(x\right)=30x$. The cost $C$, in dollars, of selling x clocks is $C\left(x\right)=0.1x^2+7x+400$.

(a) Find the profit function, $P\left(x\right)=R\left(x\right)-C\left(x\right)$.

(b) Find the profit if $x=30$ clocks are sold.

(c) Interpret $P\left(30\right)$.

Step-by-Step Solution

Answer

(a) The profit function is $P\left(x\right)=-0.1x^2+23x-400$

(b) The profit is $\$200$ if  hundred cell phones are sold.

(c) When 30 clocks are sold, the profit is $\$200$.

Step 1. Given Information

Profit Function Suppose that the revenue $R$, in dollars, from selling $x$ clocks is $R\left(x\right)=30x$. The cost $C$, in dollars, of selling $x$clocks is $C\left(x\right)=0.1x^2+7x+400$.

(a) Find the profit function, $P\left(x\right)=R\left(x\right)-C\left(x\right)$.

(b) Find the profit if $x=30$ clocks are sold.

(c) Interpret $P\left(30\right)$.

Part (a) Step 1. We have to find the profit function P ( x ) = R ( x ) - C ( x )

The value of $R\left(x\right)=30x$ and $C\left(x\right)=0.1x^2+7x+400$

Part (a) Step 2. Putting the value in the equation of profit.

$$\begin{gathered} P\left(x\right)=30x-(0.1x^2+7x+400) \\ P\left(x\right)=30x-0.1x^2-7x-400 \end{gathered}$$

Simplify 

$P\left(x\right)=-0.1x^2+23x-400$

Part (b) Step 1. We have to find the profit if x = 30 clocks phones are sold. Putting x = 30 in the function of profit.

$$\begin{gathered} P\left(30\right)=-0.1{\left(30\right)}^2+23\times 30-400 \\ P\left(30\right)=-0.1\times 900+23\times 30-400 \\ P\left(30\right)=-90+690-400 \\ P\left(30\right)=200 \end{gathered}$$

Part (c) Step 1. We have to interpret P ( 30 )

When 30 clocks are sold, the profit is $\$200$.