Question

a popular video game retailer develops apps. the total revenue, in dollars per day, after x days can be modeled by the function ( r(x) = 2x^3 + 11x^2 - 23x - 14 ). the total cost, in dollars per day, after x days, can be modeled by the function ( c(x) = 7x^2 + 15x + 26 ). after how many days does the retailer start making money?
options: 2, 1, 5, 4 (with radio buttons)

Explanation:

To determine when the retailer starts making money, we need to find when revenue \( R(x) \) exceeds cost \( C(x) \), i.e., \( R(x) - C(x) > 0 \).

Step 1: Calculate \( R(x) - C(x) \)

\[

$$\begin{align*} R(x) - C(x) &= (2x^3 + 11x^2 - 23x - 14) - (7x^2 + 15x + 26) \\ &= 2x^3 + 11x^2 - 7x^2 - 23x - 15x - 14 - 26 \\ &= 2x^3 + 4x^2 - 38x - 40 \end{align*}$$

\]

Step 2: Test each option

• For \( x = 1 \):

\[
2(1)^3 + 4(1)^2 - 38(1) - 40 = 2 + 4 - 38 - 40 = -72 < 0
\]

• For \( x = 2 \):

\[
2(8) + 4(4) - 38(2) - 40 = 16 + 16 - 76 - 40 = -84 < 0
\]

• For \( x = 4 \):

\[
2(64) + 4(16) - 38(4) - 40 = 128 + 64 - 152 - 40 = 0
\]

• For \( x = 5 \):

\[
2(125) + 4(25) - 38(5) - 40 = 250 + 100 - 190 - 40 = 120 > 0
\]

Answer:

5 (since at \( x = 5 \), \( R(x) - C(x) > 0 \), meaning revenue exceeds cost)