SOLUTION: i need this for my exam plz help me if you can Research shows that the demand function for a new product is d(x) = - 5x + 18, where x represents the number of items in thousa

Question 1169542: i need this for my exam plz help me if you can
Research shows that the demand function for a new product is d(x) = - 5x + 18, where x represents the number of items in thousands and d represents the item price in $.The cost function is C(x) = 2x + 9.
a. State the Revenue function R(x)
b. Find the corresponding Profit function P(x).
c. How many items must be sold to maximize profit?
c. How many items must be sold for the company to break even?
Hint: R(x) = x p(x). The profit function is the difference P(x) = R(x) - C(x).

Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
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.
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