The profit, P, in millions of dollars, for producing x custom built executive jet aircraft is given by P(x) = 24(x + 1)^1/2 - 3x - 24 . For what number of jets is the profit maximized?
2006-10-22 05:25:22 · 5 answers · asked by Olivia 4 in Science & Mathematics ➔ Mathematics
To find the maximum, compute
P'(x) = 24(1/2)(x+1)^(-1/2) - 3 = 12/sqrt(x+1) - 3.
Now, find x for which P'(x) = 0. Solving this equation for x gives
12/sqrt(x+1) = 3, so 4 = sqrt(x+1) => 16 = x+1, so x=15.
You can confirm that this critical number corresponds to a local maximum of P, rather than a local minimum, by computing P''(15), which will turn out to be negative, so it is a local maximum.
This is also the absolute maximum, because you must require x>= 0, and P(0) is negative. Also, as x -> infinity, P(x) -> -infinity, because 3x grows more rapidly than 24(x+1)^(1/2). Therefore the maximum occurs at x=15.
2006-10-22 05:32:08 · answer #1 · answered by James L 5 · 1⤊ 0⤋
the first differential of P(x)
=12/(x+1)^(1/2)-3
the second differential is -ve
hence P(x) is max at P'(x)=0
therefore,
12/(x+1)^(1/2)-3=0
(x+1)^(1/2)=4
>>>>>>x=15 at max profit
therefore, the profit is maximised when
15 jets are built
2006-10-22 12:56:58 · answer #2 · answered by Anonymous · 0⤊ 0⤋
Well being P(x) = 24(x+1)^(1/2)-3x-24
P`(x) = (12/(x+1))-3
This function takes 0 as its value when x is = to 15
2006-10-22 12:37:46 · answer #3 · answered by Guillermo I 1 · 1⤊ 0⤋
No, I think the ans is 15.
2006-10-22 12:29:02 · answer #4 · answered by Aditya 1 · 1⤊ 0⤋
just take the derivative dP/dx and solve for x = 0.
2006-10-22 12:36:45 · answer #5 · answered by MrZ 6 · 0⤊ 1⤋