A one-product company finds that its profit, P in millions of dollars, is given

Question

A one-product company finds that its profit, P in millions of dollars, is given by the following equation where a is the amount on advertising in million of dollars, and p is the price charged per item of the product in dollars:

P(a,p)= 4ap+50p-9p^2-1/10a^2p-120

Find the maximum value of P and the values of a and p at which it is attained.

A) The maximum of P is attained when a is $____ million and p is $___


B)The maximum value of P is $___ Million

What is the answer for A and what is the answer for B in millions? Please

Explanation / Answer

Pa = 4p - 1(-2)a-3/(10p) = 0

==> 4pa3 + 2/10p =0

==> 4p2a3 = -1/5

==>(a3p2) = -1/20 =============> (ap)2a = -1/20 ---> (1)

Pp = 4a + 50 -18p -(1/10a2)(-1/p2) = 0

==> 8 a3p2 + 50 (ap)2 - 18 p(ap)2 = 1/10

==> -8/20 + 50(-1/(20a)) - 18p(-1/(20a)) = 1/10

==> -0.4 -2.5/a + 0.9p/a = 0.1

==> [0.9p-2.5]/a = 0.5

==> 0.9p -2.5 =0.5a

==> (1.8)(p/a) - 5/a = 1

1.8p -5 =a==> 1.8p = (a+5) ==> p = (a+5)/1.8

we know (a3p2) = -1/20

==> a3 [ (a+5)/1.8]2 = -1/20

==> a3(a2+10a+25) = -3.24/20

==> a5 + 10a4 + 25a3 = -0.162

from this find a, then substitue in the (1) and get p

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