Question

Find the demand function for the marginal revenue function. Recall that if no items are sold, the revenue is 0. R’(x)=0.06x^2-0.07x+155.
Find p(x).

Answer 1

To find the demand function, we need to integrate the marginal revenue function R'(x) to get the revenue function R(x), and then differentiate the revenue function to get the demand function.

Given that R'(x) = 0.06x^2 - 0.07x + 155, we integrate this function to find R(x):

R(x) = ∫(0.06x^2 - 0.07x + 155) dx
R(x) = 0.02x^3 - 0.035x^2 + 155x + C

Since revenue is 0 when no items are sold, we can find the value of the constant by setting R(0) = 0:

0 = 0.02(0)^3 - 0.035(0)^2 + 155(0) + C
C = 0

So, the revenue function is R(x) = 0.02x^3 - 0.035x^2 + 155x.

To find the demand function p(x), we differentiate the revenue function R(x):

p(x) = R'(x) = dR(x)/dx
p(x) = 0.06(3)x^2 - 0.07(2)x + 155
p(x) = 0.18x^2 - 0.14x + 155

Therefore, the demand function is p(x) = 0.18x^2 - 0.14x + 155.