The revenue and cost functions for the housing developer are:
C(n) = 8 + 0.065n
R(n) = 1.6 √n
Suppose that the developer found a way to reduce her variable cost to $58 000 per house. How would this affect:
i) the minimum and maximum number of houses she could build?
ii) her maximum potential profit?
To analyze how the reduction in variable cost to $58,000 per house would affect the minimum and maximum number of houses the developer could build and her maximum potential profit, we need to understand the relationships between the cost (C), revenue (R), and profit (P) functions.
The profit (P) function is given by subtracting the cost function (C) from the revenue function (R):
P(n) = R(n) - C(n)
i) To find the minimum and maximum number of houses the developer could build, we need to determine the points at which the profit function reaches extreme values.
To find the minimum, we need to find the critical points where the derivative of the profit function is equal to zero:
P'(n) = 0
Let's calculate the derivative of the profit function:
P'(n) = R'(n) - C'(n)
The derivative of the revenue function R(n) with respect to n is given by:
R'(n) = (d/dn) [1.6 √n] = (1/2) * (1.6) * (n^(-0.5)) = 0.8 / √n
The derivative of the cost function C(n) with respect to n is a constant, as the variable cost per house is fixed and independent of the number of houses built:
C'(n) = 0
Now, we set P'(n) = 0:
0.8 / √n - 0 = 0
Solving this equation, we find:
0.8 / √n = 0
Since we cannot divide by zero, this equation has no solution. Therefore, there are no critical points, and the developer can build any number of houses she desires restricted only by practical considerations.
ii) To find the developer's maximum potential profit, we can either scan the profit function or identify the highest point on the graph of the profit function. However, given that we are not provided with additional information or constraints, we cannot determine the maximum potential profit solely based on the given functions.
To find the maximum potential profit, we need additional information, such as the total fixed costs or a constraint defining the maximum number of houses the developer can build. Then, we can optimize the profit function within the given constraints using techniques like optimization or constrained optimization algorithms.