The unit selling price p (in dollars) and the quantity demanded x (in pairs) of a certain brand of women’s shoes are given by the demand equation

P(x) = 100e^-0.001x f or 0 _< x _< 20,000.

a. Find the revenue function, R. (Hint: R(x) = x(p(x)), since the revenue function is the unit selling price at a demand level of x units times the number of x units demanded.)

b. Find the marginal revenue function, R’.

c. What is the marginal revenue when, x = 10 ?

a. To find the revenue function, R(x), we need to multiply the unit selling price, p(x), by the quantity demanded, x.

Given that p(x) = 100e^(-0.001x), we can substitute this into the revenue function equation:

R(x) = x * p(x)
= x * (100e^(-0.001x))

Therefore, the revenue function is R(x) = 100xe^(-0.001x).

b. The marginal revenue function, R'(x), represents the rate of change of revenue with respect to changes in the quantity demanded. We can find it by taking the derivative of the revenue function R(x) with respect to x:

R'(x) = dR(x)/dx

To find this derivative, we need to apply the product rule. Let's break down the terms to make it easier:

R(x) = 100xe^(-0.001x)
= 100x * e^(-0.001x)

Now, applying the product rule:

R'(x) = 100 * e^(-0.001x) + 100x * (-0.001) * e^(-0.001x)

Simplifying further, we can factor out e^(-0.001x):

R'(x) = e^(-0.001x) * (100 - 0.001x * 100)

Therefore, the marginal revenue function is R'(x) = e^(-0.001x) * (100 - 0.001x).

c. To find the marginal revenue when x = 10, we can substitute x = 10 into the marginal revenue function:

R'(10) = e^(-0.001 * 10) * (100 - 0.001 * 10)

Using a calculator to evaluate e^(-0.001 * 10), we get approximately e^(-0.01) = 0.99004983375.

Substituting this into the equation:

R'(10) = 0.99004983375 * (100 - 0.001 * 10)

Calculating the expression in the brackets:

R'(10) = 0.99004983375 * (100 - 0.01)

Finally, solving the calculation:

R'(10) ≈ 98.004983375

Therefore, the marginal revenue when x = 10 is approximately 98.004983375 dollars.