Question

Hello! Can u help me with this please. I'm having a hard time solving this one but I don't know if I make it right. Please help me.

Suppose that 10,000𝑥 dollars is the inventory carried in a store
employing 𝑦 clerks, 𝑃 dollars is the weekly profit of the store, and
𝑃 = 3000 + 240𝑦 + 20𝑦 ( 𝑥 − 2𝑦 ) − 10(𝑥 − 12)²
where 15 ≤ 𝑥 ≤ 25 and 5 ≤ 𝑦 ≤ 12 . At present the inventory is
$180,000 and there are 8 clerks.


(a) Find the instantaneous rate of change of 𝑃 per unit change in 𝑥 if
𝑦 remains fixed at 8.
(b) Use the result of part (a) to find the approximate change in the
weekly profit if the inventory changes from $180,000 to $200,000 and the number of clerks remains fixed at 8.
(c) Find the instantaneous rate of change of 𝑃 per unit change in 𝑦 if 𝑥 remains fixed at 18.
(d) Use the result of part (c) to find the approximate change in the
weekly profit if the number of clerks is increased from 8 to 10 and the inventory remains fixed at $180,000.

Answer 1

p = 3000 + 240y + 20y (x-2y) - 10(x-12)^2
(a) dp/dx = 20*8x - 20(x-12)

(b) ∆p ≈ dp/dx ∆x
so, when the inventory changes from 180K to 200K, x changes from 18 to 20
∆p ≈ (160*18 - 20(18-12)) * (20-18) = ____

(c) p = 3000 + 240y + 20y (x-2y) - 10(x-12)^2
dp/dy = 240 + 20(18-2y) + 20y(-2)

(d) do as in (c)

Answer 2

Thank u so much. It means a lot.