help me to do this math question about derivative?

Question

.A fried chicken franchise finds that the demand equation for its new roast chicken product, "Roasted Rooster", is given by
p=48/ q^1.5
where p is the price (in dollars) per quarter-chicken serving and q is the number of quarter-chicken servings that can be sold per hour at this price. Express q as a function of p, and find the elasticity of demand when the price is set at $6 per serving.

Answer 1

p * q^1.5 = 48
q^1.5 = 48 / p
q = (48 / p)^(2/3) = (48 * p^-1)^(2/3)

When p = 6,

q = 8^(2/3) = 4

As a check,

6 = 48 / 4^1.5
6 = 48 / 8

Yes!!!

BabyGirl is wrong in her third equation. She handles the 1.5 incorrectly.

The elasticity of demand at any particular price is (% Change in Quantity Demanded)/(% Change in Price).

Let us start by taking the first derivative of the demand function:

q' = (2/3)(48 * p^-1)^(-1/3) * (-48) * (p^-2)

For p = 6,

q' = (2/3) * 8^(-1/3) * (-48) * (6^-2)
= (2/3) * (1/2) * (-48) * (1/36)
= (1/3) * (-48) * (1/36)
= -16 * 1/36
= -16/36 = -4/9

This means that at p = 6, for each dollar increase in p, the quantity demanded goes down by 4/9.

Let's do a rough check on that before going on.

For p = 7,

q = (48/7)^(2/3) = 3.6, down from 4.

Close enough. Let's go on.

The % change in price is (7 - 6) / 6 = 1/6
The % change in the quanitity demanded is

(old quantity - new quantity) / old quantity
(4/9) / 4 = 1/9

The elasticity of demand is

(1/9) / (1/6) = 2/3

An elasticity of less than 1 means that as the price increases, the quantity demanded does not fall proportionately. It falls less than proportionately, leading to an increase in gross revenue.

Let's see. At p = 6, the quantity demanded was 4, for a gross revenue of 24. At p = 7, the quantity demanded is 4 - 4/9, or 32/9, leading to a gross revenue of 7 * 32/9 = 224/9 = 24.89, up from 24.

Answer 2

I'm not sure, but here goes...

q^1.5 = 48/p

1.5 ln q = ln (48/p)

q = {e^(48/p} / 1.5

When p =6

q = e^8/1.5
q' =