Question

Given the following price-demand and cost functions:

p(x) = 27-3x, C(x) = 15x + 9,
where p(x) is the wholesale price in dollars at which a thousand units of product
can be sold and C(z) is in thousands of dollars, both functions have the domain
0≤x≤9,
(A) Form the revenue function R(x) and find the break-even points;
(B) Form the profit function P(x) and find the maximum profit;
(C) What price will maximize the profit?

Answer 1

To answer this question, we need to form the revenue function, find the break-even points, form the profit function, find the maximum profit, and determine the price that maximizes the profit.

Step-by-step explanation:

Forming the Revenue Function

The revenue function R(x) can be obtained by multiplying the price function p(x) by the quantity x:

R(x) = p(x) * x = (27-3x) * x

Finding the Break-Even Points

The break-even points occur when the revenue is equal to the cost. Set R(x) equal to the cost function C(x) and solve for x:

(27-3x) * x = 15x + 9

Solve this equation to find the break-even points.

Forming the Profit Function

The profit function P(x) can be obtained by subtracting the cost function C(x) from the revenue function R(x):

P(x) = R(x) - C(x) = (27-3x) * x - (15x + 9)

Finding the Maximum Profit

To find the maximum profit, we need to find the value of x that maximizes the profit function P(x). Take the derivative of P(x) with respect to x and set it equal to zero. Solve the resulting equation to find the value of x that maximizes the profit.

Price that Maximizes Profit

To find the price that maximizes the profit, substitute the value of x that maximizes the profit into the price function p(x). This will give you the price at which the maximum profit occurs.

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