Question

Profit Function Suppose that the revenue $R$, in dollars, from selling x cell phones, in hundreds, is $R\left(x\right)=-1.2x^2+220x$. The cost $C$, in dollars, of selling x cell phones is $C\left(x\right)=0.05x^3-2x^2+65x+500$.

(a) Find the profit function, role="math" localid="1646408711167" $P\left(x\right)=R\left(x\right)-C\left(x\right)$.

(b) Find the profit if $x=15$hundred cell phones are sold.

(c) Interpret $P\left(15\right)$.

Short answer

(a) The profit function is $P\left(x\right)=-0.05x^3+0.8x^2+155x-500$

(b) The profit is $\$1836.25$if $x=15$hundred cell phones are sold.

(c) When 15 hundred cellphones are sold, the profit is$\$1836.25$.

Step-by-step solution

Step 1. Given Information

Profit Function Suppose that the revenue $R$, in dollars, from selling x cell phones, in hundreds, is role="math" localid="1646408800086" $R\left(x\right)=-1.2x^2+220x$. The cost $C$, in dollars, of selling x cell phones is role="math" localid="1646408922328" $C\left(x\right)=0.05x^3-2x^2+65x+500$.

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

(b) Find the profit if $x=15$hundred cell phones are sold.

(c) Interpret $P\left(15\right)$.

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

The value of$R\left(x\right)=-1.2x^2+220x$and$C\left(x\right)=0.05x^3-2x^2+65x+500$

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

$$\begin{gathered} P\left(x\right)=-1.2x^2+220x-(0.05x^3-2x^2+65x+500) \\ P\left(x\right)=-1.2x^2+220x-0.05x^3+2x^2-65x-500 \end{gathered}$$

Simplify

role="math" localid="1646409621483" $P\left(x\right)=-0.05x^3+0.8x^2+155x-500$

Part (b) Step 1. We have to find the profit if x=15 hundred cell phones are sold.

Putting$x=15$in the function of profit.

role="math" localid="1646409665519" $$\begin{gathered} P\left(15\right)=-0.05{\left(15\right)}^3+0.8{\left(15\right)}^2+155\times 15-500 \\ P\left(15\right)=-0.05\times 3375+0.8\times 225+155\times 15-500 \\ P\left(15\right)=-168.75+180+2325-500 \\ P\left(15\right)=1836.25 \end{gathered}$$

Part (c) Step 1. We have to interpret P(15).

When 15 hundred cellphones are sold, the profit is$\$1836.25$.