Chapter 2: Q. 103 (page 71)

Profit Function Suppose that the revenue $R$ , in dollars, from selling x cell phones, in hundreds, is $R (x) = - 1.2 x^{2} + 220 x$ . The cost $C$ , in dollars, of selling x cell phones is $C (x) = 0.05 x^{3} - 2 x^{2} + 65 x + 500$ .
(a) Find the profit function, role="math" localid="1646408711167" $P (x) = R (x) - C (x)$ .
(b) Find the profit if $x = 15$ hundred cell phones are sold.
(c) Interpret $P (15)$ .

Short Answer

Expert verified

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

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

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" $R (x) = - 1.2 x^{2} + 220 x$ . The cost $C$ , in dollars, of selling x cell phones is role="math" localid="1646408922328" $C (x) = 0.05 x^{3} - 2 x^{2} + 65 x + 500$ .

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

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

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

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

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

$P (x) = - 1.2 x^{2} + 220 x - (0.05 x^{3} - 2 x^{2} + 65 x + 500) P (x) = - 1.2 x^{2} + 220 x - 0.05 x^{3} + 2 x^{2} - 65 x - 500$

Simplify

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

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"

P (15) = - 0.05 {(15)}^{3} + 0.8 {(15)}^{2} + 155 \times 15 - 500 P (15) = - 0.05 \times 3375 + 0.8 \times 225 + 155 \times 15 - 500 P (15) = - 168.75 + 180 + 2325 - 500 P (15) = 1836.25

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

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

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Short Answer

Step by step solution

Step 1. Given Information

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

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

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

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

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