Assume the only choice variable is total benefit function is B(x) = 170x-x², and cost function is C(x) = 100-10x + 2x².
a. What are the marginal benefit and marginal cost functions?
b. Set up the net benefit function and then determine the level x that maximises net benefit.
c. what is the maximum level of net benefit?
gjkl5563
answer
a. To find the marginal benefit function, we need to take the derivative of the total benefit function with respect to x:
B'(x) = d/dx (170x - x^2)
Using the power rule, we get:
B'(x) = 170 - 2x
So, the marginal benefit function is B'(x) = 170 - 2x.
To find the marginal cost function, we need to take the derivative of the cost function with respect to x:
C'(x) = d/dx (100 - 10x + 2x^2)
Using the power rule, we get:
C'(x) = -10 + 4x
So, the marginal cost function is C'(x) = -10 + 4x.
b. The net benefit function is the difference between the total benefit function and the cost function. So, we have:
N(x) = B(x) - C(x)
= (170x - x^2) - (100 - 10x + 2x^2)
= 170x - x^2 - 100 + 10x - 2x^2
Simplifying, we get:
N(x) = -3x^2 + 180x - 100
To determine the level x that maximizes net benefit, we need to find the critical points of the net benefit function. We can do this by finding where the derivative of N(x) is equal to zero:
N'(x) = d/dx (-3x^2 + 180x - 100)
Using the power rule, we get:
N'(x) = -6x + 180
Setting N'(x) equal to zero and solving for x:
-6x + 180 = 0
6x = 180
x = 30
So, the level x that maximizes net benefit is x = 30.
c. To find the maximum level of net benefit, we substitute the value x = 30 into the net benefit function:
N(30) = -3(30)^2 + 180(30) - 100
= -3(900) + 5400 - 100
= -2700 + 5400 - 100
= 2700
Therefore, the maximum level of net benefit is 2700.