Question

An analyst has found that a company's costs and revenues in dollars. for a particular product are given by

C(x)=2x and R(x)=6x− x^2/1000.
Find the maximum profit when sales are between 1000 and 4000 items.

Answer 1

To find the maximum profit, we need to determine the level of sales that maximizes the difference between revenue and cost. The profit function P(x) can be expressed as the difference between the revenue function R(x) and the cost function C(x):

P(x) = R(x) - C(x)

Given that C(x) = 2x and R(x) = 6x - x^2/1000, we can substitute these expressions into the profit function:

P(x) = (6x - x^2/1000) - 2x

Simplifying this expression, we have:

P(x) = 4x - x^2/1000

The next step is to find the value of x that maximizes the profit function P(x), within the range of sales between 1000 and 4000 items. To do this, we can take the derivative of P(x) with respect to x and set it equal to zero:

P'(x) = 4 - (2x/1000) = 0

Solving this equation for x, we get:

4 - (2x/1000) = 0

2x/1000 = 4

2x = 4000

x = 2000

So, within the given range of sales, the value of x that maximizes profit is 2000 items.

To find the maximum profit, we can substitute this value of x back into the profit function:

P(x) = 4x - x^2/1000

P(2000) = 4(2000) - (2000^2)/1000

P(2000) = 8000 - 4000000/1000

P(2000) = 8000 - 4000

P(2000) = 4000

Therefore, the maximum profit when sales are between 1000 and 4000 items is $4000.

Answer 2

To find the maximum profit when sales are between 1000 and 4000 items, we need to determine the number of items that will maximize the profit.

The profit function is given by P(x) = R(x) - C(x), where R(x) represents the revenue and C(x) represents the cost.

1. Start by substituting the given equations for C(x) and R(x) into the profit function:

P(x) = (6x - x^2/1000) - 2x.

2. Simplify the expression:

P(x) = 6x - x^2/1000 - 2x.

3. Combine like terms:

P(x) = 4x - x^2/1000.

4. To find the maximum profit, we need to find the vertex of the parabolic function. The vertex can be found using the formula x = -b/2a, where a and b are the coefficients of the quadratic equation.

5. In this case, the quadratic equation is -x^2/1000 + 4x. Therefore, a = -1/1000 and b = 4.

6. Substitute the values of a and b into the formula to find the x-coordinate of the vertex:

x = -4 / (2 * (-1/1000)).

7. Simplify the expression:

x = -4000.

8. Since we are considering sales between 1000 and 4000 items, we can ignore the negative value of x and consider x = 4000.

9. Substitute x = 4000 back into the profit function to find the maximum profit:

P(4000) = 4(4000) - (4000^2)/1000.

10. Simplify the expression:

P(4000) = 16000 - 16000000/1000.

11. Calculate the value of P(4000):

P(4000) = 16000 - 16000.

12. The maximum profit is zero, which means that the company breaks even when sales are between 1000 and 4000 items.

Therefore, the maximum profit when sales are between 1000 and 4000 items is zero.

Step-by-step explanation: