Question 1169542
Absolutely! Let's solve this problem step-by-step.

Understanding the Problem

We are given the demand function d(x) and the cost function C(x). We need to find the revenue function, profit function, the number of items to maximize profit, and the number of items to break even.

Given Information

Demand function: d(x) = -5x + 18 (price per item)
Cost function: C(x) = 2x + 9
a) Revenue Function R(x)

Revenue is the product of the number of items sold and the price per item.
R(x) = x * d(x)
R(x) = x * (-5x + 18)
R(x) = -5x² + 18x
b) Profit Function P(x)

Profit is the difference between revenue and cost.
P(x) = R(x) - C(x)
P(x) = (-5x² + 18x) - (2x + 9)
P(x) = -5x² + 18x - 2x - 9
P(x) = -5x² + 16x - 9
c) Number of Items to Maximize Profit

To maximize profit, we need to find the vertex of the profit function P(x), which is a quadratic function.

The x-coordinate of the vertex is given by x = -b / (2a), where P(x) = ax² + bx + c.

In our case, a = -5 and b = 16.

x = -16 / (2 * -5) = -16 / -10 = 1.6

Since x represents the number of items in thousands, we need to sell 1.6 * 1000 = 1600 items to maximize profit.

d) Number of Items to Break Even

To break even, the profit must be zero (P(x) = 0).

We need to solve the equation -5x² + 16x - 9 = 0.

We can use the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a)

In our case, a = -5, b = 16, and c = -9.

x = (-16 ± √(16² - 4 * -5 * -9)) / (2 * -5)

x = (-16 ± √(256 - 180)) / -10

x = (-16 ± √76) / -10

x = (-16 ± 8.7178) / -10

x1 = (-16 + 8.7178) / -10 = -7.2822 / -10 = 0.72822

x2 = (-16 - 8.7178) / -10 = -24.7178 / -10 = 2.47178

Since x represents thousands of items, we have:

x1 ≈ 0.72822 * 1000 ≈ 728 items
x2 ≈ 2.47178 * 1000 ≈ 2472 items
Therefore, the company breaks even when selling approximately 728 items or 2472 items.

Final Answers

a) R(x) = -5x² + 18x
b) P(x) = -5x² + 16x - 9
c) 1600 items
d) Approximately 728 items or 2472 items.