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?
Is n the number of houses sold? What are the units of C and R? Dollars or thousands of dollars? Why should R be proportional to √n and not n?
Why should there be a minimum number built (other than zero)?
I can't make sense of your question.
To determine how the reduction in variable cost to $58,000 per house would affect the minimum and maximum number of houses the developer could build, as well as her maximum potential profit, we can use the given revenue and cost functions.
i) Minimum and Maximum Number of Houses:
The minimum and maximum number of houses the developer could build will be determined by the intersection points of the revenue (R) and cost (C) functions. At these points, the revenue earned from selling the houses will be equal to the cost incurred in building them.
Setting R(n) = C(n), we have:
1.6 √n = 8 + 0.065n
To solve this equation, we need to square both sides to eliminate the square root:
(1.6 √n)² = (8 + 0.065n)²
2.56n = 64 + 1.04n + 0.004225n²
0.004225n² + 0.976n - 64 = 0
Solving this quadratic equation will give us two possible values for n, which represent the minimum and maximum number of houses the developer could build.
ii) Maximum Potential Profit:
The maximum potential profit can be calculated by subtracting the cost function from the revenue function, considering the number of houses built.
Profit (P) = R(n) - C(n)
P(n) = (1.6 √n) - (8 + 0.065n)
To find the maximum potential profit, we need to find the value of n that maximizes the profit function.
To summarize, here's what needs to be done:
1. Solve the quadratic equation 0.004225n² + 0.976n - 64 = 0 to find the minimum and maximum number of houses the developer could build.
2. Calculate the profit function P(n) = (1.6 √n) - (8 + 0.065n).
3. Plug in the values obtained from step 1 to determine the maximum potential profit.
Please note that the calculation steps may require a calculator or software program capable of solving quadratic equations.