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 ?

Given :

R(x)= x P(x)
= x (100 e^(-.001x) )

R '(x) = x(-.001)(100) e^(-.001x) + 100 e^(-.001x)
=- .1x e^(-.001x) + 100 e(-.001x)
R' (10) = 1e^(-.01) + 100 e^(-.01)
= .....
you do the button pushing

After doing the button pushing the answer is 99.995 is that correct?

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

R(x) = x * p(x)

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

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

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

b. The marginal revenue, R', represents the rate of change of the revenue function with respect to x. To find it, we differentiate the revenue function with respect to x.

R'(x) = dR/dx

Using the product rule, we differentiate the function R(x) = 100xe^(-0.001x).

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

Simplifying further, we get:

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

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

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

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

Simplifying further, we get:

R'(10) = 100e^(-0.01) - 0.01e^(-0.01)

Using a calculator for the numerical value, we find the marginal revenue when x = 10.

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

Since p(x) is given by the equation P(x) = 100e^(-0.001x), we can substitute this into the revenue equation to get:

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

So, the revenue function is R(x) = 100xe^(-0.001x) for 0 ≤ x ≤ 20,000.

b. To find the marginal revenue function, R'(x), we need to take the derivative of the revenue function, R(x), with respect to x.

Using the product rule, we have:

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

Simplifying further, we have:

R'(x) = 100e^(-0.001x) - 0.1xe^(-0.001x)

So, the marginal revenue function is R'(x) = 100e^(-0.001x) - 0.1xe^(-0.001x).

c. To find the marginal revenue when x = 10, we can substitute x = 10 into the marginal revenue function, R'(x), and calculate the result.

R'(10) = 100e^(-0.001(10)) - 0.1(10)e^(-0.001(10))

Simplifying further, we have:

R'(10) = 100e^(-0.01) - 1e^(-0.01)

You can use a calculator or a math software to evaluate this expression.

Note: The values of R(x) and R'(x) are dependent on the value of x, so you may get different results depending on the exact values you substitute.