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 $P\left(30\right)$.

Short 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-by-step solution

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 role="math" localid="1646411590420" $C\left(x\right)=0.1x^2+7x+400$.

(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

role="math" localid="1646411303541" $P\left(x\right)=-0.1x^2+23x-400$

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 $\$200$.