Question

profit is the difference between revenue and cost. the revenue, in dollars, of a company that manufactures cell phones can be modeled by the polynomial $2x^2 + 55x + 10$. the cost, in dollars, of producing the cell phones can be modeled by $2x^2 - 15x - 40$. the variable $x$ represents the number of cell phones sold. what expression represents the profit, and what is the profit if 240 cell phones are sold? \bigcirc $40x - 30$; $\\$2,400$ \bigcirc $40x - 30$; $\\$9,570$ \bigcirc $70x + 50$; $\\$16,850$ \bigcirc $70x + 50$; $\\$28,800$

Explanation:

Step1: Recall Profit Formula

Profit \( P \) is revenue \( R \) minus cost \( C \), so \( P = R - C \).
Given \( R = 2x^2 + 55x + 10 \) and \( C = 2x^2 - 15x - 40 \).

Step2: Substitute R and C into Profit Formula

\( P=(2x^2 + 55x + 10)-(2x^2 - 15x - 40) \)
Remove parentheses: \( P = 2x^2 + 55x + 10 - 2x^2 + 15x + 40 \)

Step3: Simplify the Expression

Combine like terms:

• \( 2x^2 - 2x^2 = 0 \)
• \( 55x + 15x = 70x \)
• \( 10 + 40 = 50 \)

So, \( P = 70x + 50 \)

Step4: Calculate Profit for x = 240

Substitute \( x = 240 \) into \( P = 70x + 50 \):
\( P = 70(240)+50 \)
\( 70\times240 = 16800 \), then \( 16800 + 50 = 16850 \)? Wait, no, wait: Wait, 70240: 70200=14000, 7040=2800, 14000+2800=16800. Then 16800 + 50 = 16850? Wait, no, wait the options: Wait, wait, maybe I made a mistake. Wait, let's re - check the revenue and cost polynomials. Wait, the revenue is \( 2x^2 + 55x + 10 \), cost is \( 2x^2 - 15x - 40 \). Wait, when we subtract: \( (2x^2 + 55x + 10)-(2x^2 - 15x - 40)=2x^2 + 55x + 10 - 2x^2 + 15x + 40 = 70x + 50 \). Then for x = 240: 70240=16800, 16800 + 50 = 16850. Wait, but let's check the options. The third option is \( 70x + 50; \$16,850 \). Wait, but wait, maybe I misread the revenue or cost. Wait, the revenue is \( 2x^2 + 55x + 10 \), cost is \( 2x^2 - 15x - 40 \). Yes, the subtraction is correct. So the profit expression is \( 70x + 50 \), and when x = 240, profit is \( 70*240 + 50=16800 + 50 = 16850 \). Wait, but let's check the options again. The third option is \( 70x + 50; \$16,850 \), the fourth is \( 70x + 50; \$28,800 \). Wait, no, 70240 is 16800, plus 50 is 16850. So the correct option is the one with \( 70x + 50 \) and \( \$16,850 \). Wait, but let's re - calculate 70240: 24070. 2407 = 1680, so 240*70 = 16800. Then 16800 + 50 = 16850. So the profit expression is \( 70x + 50 \), and the profit when 240 are sold is \( \$16,850 \).

Wait, but earlier I thought maybe a mistake, but the calculation seems correct. So the correct option is the third one: \( 70x + 50; \$16,850 \)

Answer:

The expression for profit is \( 70x + 50 \), and the profit when 240 cell phones are sold is \( \$16,850 \). So the correct option is C. \( 70x + 50; \$16,850 \) (assuming the options are labeled A, B, C, D with C being \( 70x + 50; \$16,850 \))