Per Case

The consumer demand curve for cases of gadgets is given by q=(90-p)^2, where p is the price per case and q ?

is the demand in weekly sells.
a- If the gadgets cost $30 per case to produce, use calculus to determine the price that should be charged per case to maximize the weekly profit. The answer is $50. How do you find this?
b-Determine the elasticity of the demand if the price is set at $35 per case. Interpret the elasticity in the context of the problem.
The answer is E=1.27 with a line over it. How do you find this?

Part a

Sales volume q = (90 – p)^2
Revenue R = p x q
Cost of goods sold C = 30 x q

Profit P = R – C
P = p x q -30 x q = q x (p-30)

P = (90-p)^2 x (p-30)
P = (8100 -180p + p^2) x (p – 30)
P = p^3 – 210p^2 + 13500p – 243000

p at maximum profit is found by solving dP/dp = 0

dP/dp = 3p^2 -420p + 13500 = 0

Solutions for dP/dp = 0 are p =50 and p = 90
p = 90 results in q = 0, so p = 50 is the solution of interest

Part b added a little later:

Price elasticity of demand e = dq/dp x p/q

q = p^2 -180p + 8100

dq/dp = 2p -180

p = 35, q = 3025

e = -1.27

I don’t know why you have the answer as an absolute value. The negative 1.27 value indicates that if the price is raised by 1% above $35, the demand quantity will drop by 1.27%.

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