A property developer is planning to build houses on 4 acres of land. She estimates that if she builds 16 houses, the profit per house will be $400,000, and that each additional house she builds will reduce her profit per house by $10,000. The total profit P(n), in thousands of dollars, from making n houses, can be modelled by the function:
(a) How much profit will the developer make from making 9 houses according to thisCode:P(n)= 560n-10n^2
$4, 230, 000
(b) Find the roots of the equation P(n) = 0, and use these to find how many houses the developer should build for maximum profit. State the value of this profit.
The roots are when n = 0 or 56
build 28 houses for maximum profit
$7,840,000 is the value of the profit if the developer builds 28 houses
(c) What is a sensible practical domain for P(n)?
starts with n = 0 – can’t have negative number of houses and ends with n= 28 – as this is the maximum value of P(n), after this the profit starts decreasing.
(d) The developer needs to make at least $6,800,000 (i.e. $6800 thousand) to invest in a block of land she wishes to purchase for her next development project. Find algebraically the roots of the equation P(n) = 6800, and use your answer to suggest how many houses she needs to build to gain this much profit.Code:0<= n <= 28
n = 17.80196097 or 38.19803903Code:560n-10n^2 -6800 = 0
She should build 38 houses to make at least $6,800,000
I'm sorry I didn't include my working, but its quiet difficult to type it all out. Any assistance with this would be great, thanks for looking