To solve this problem, let's break it down step by step.
1. Understanding the Problem:
- We are given the revenue function: [tex]\( R(x) = 2x^2 + 55x + 10 \)[/tex].
- We are also given the cost function: [tex]\( C(x) = 2x^2 - 15x - 40 \)[/tex].
- We need to find the profit function, [tex]\( P(x) \)[/tex], which is the difference between the revenue and the cost: [tex]\( P(x) = R(x) - C(x) \)[/tex].
2. Finding the Profit Function:
- The profit function is obtained by subtracting the cost function from the revenue function:
[tex]\[
P(x) = (2x^2 + 55x + 10) - (2x^2 - 15x - 40)
\][/tex]
- We now distribute the negative sign through the cost function:
[tex]\[
P(x) = 2x^2 + 55x + 10 - 2x^2 + 15x + 40
\][/tex]
- Combine like terms:
[tex]\[
P(x) = (2x^2 - 2x^2) + (55x + 15x) + (10 + 40)
\][/tex]
[tex]\[
P(x) = 0 + 70x + 50
\][/tex]
- Thus, the profit function simplifies to:
[tex]\[
P(x) = 70x + 50
\][/tex]
3. Calculating the Profit for 240 Cell Phones:
- To find the profit when [tex]\( x = 240 \)[/tex], we substitute [tex]\( x = 240 \)[/tex] into the profit function:
[tex]\[
P(240) = 70 \cdot 240 + 50
\][/tex]
- Simplify the expression:
[tex]\[
P(240) = 16800 + 50
\][/tex]
[tex]\[
P(240) = 16850
\][/tex]
4. Result:
- The profit function is [tex]\( 70x + 50 \)[/tex].
- The profit when 240 cell phones are sold is [tex]\( \$16850 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{70 x+50 ; \$ 16,850} \][/tex]
