Question

A person wants to plant some trees in his community park. The local nursery has to perform this task. It charges the cost of planting trees by the following formula:
$C\left(x\right)=x^3-45x^2+600x,$ where x is the number of trees and C(x) is the cost of planting x trees in rupees. The local authority has imposed a restriction that it can plant 10 to 20 trees in one community park for a fair distribution. For how many trees should the person place the order so that he has to spend the least amount? How much is the least amount? Use calculus to answer these questions. Which value is being exhibited by the person?

Answer 1

We've $C^{\prime }\left(x\right)=3x^2-90x+600=3(x-10)(x-20),C"\left(x\right)=6x-90$
For the time being we may assume that the function C(x) is continuous at all the points in the interval [10, 20].
Now $\text{C'(x) = 3x}{}^2\text{- 90x + 600 = 3(x-10)(x-20), C"(x) = 6x - 90}$
For $=C^{\prime }\left(x\right)=3(x-10)(x-20)=0\Rightarrow x=10,20.$
Note that C"(10) = - 30 < 0 and C"(20) = 30 > 0.
So C(x) is minimum at x = 20 and maximum at x = 10.
Hence C(10) = 2500, C(20) = 2000.
Therefore the person must place the order for 20 trees in order to spend the least amount which is Rs 2000.