The pizzas at a restaurant are circular and the price of a pizza depends on the
ID: 2889845 • Letter: T
Question
The pizzas at a restaurant are circular and the price of a pizza depends on the diameter of the pizza. Recall that the area A of a circle of radius r is given by A = *r^2.
Normally, one expects the cost of the pizza per unit area to decrease as the size of the pizza increases. That is, you expect to get more pizza for your money if you buy a larger pizza. A local pizza chain offers the following prices for a non-discounted cheese pizza. (The following measurements refer to the diameter of the pizza.)
10-inch $ 8.00
12-inch $10.00
14-inch $12.00
16-inch $14.00
Let P denote the price of the pizza and let d denote the diameter of the pizza. A simple model for the price of a pizza would involve fixed costs (salary of employees, rental for the restaurant, insurance, utility costs, etc.) and the cost to make each pizza (dough, cheese, tomato sauce, etc.). The cost of the ingredients for a single pizza certainly depends on the size of the pizza.
(a) Given this model, why is it reasonable to consider setting the price of a pizza to be P (d) = a*d^2 + b where a and b are real numbers to be determined.
(b) How do the fixed costs appear in this formula? How does the cost of the ingredients of the pizza appear in this formula?
(c) Consider the function P (d) = .04*d^2 + 4. Compute P (10), P (12), P (14), P (16) and compare with the prices above.
(d) Use the formula P (d) = .04*d^2 + 4 to compute the formula for the price per unit area.
(e) Explain how the algebraic expression in (d) shows that the price per unit area is de- creasing as the size of the pizza increases.
Explanation / Answer
From the given question,
a) price of pizza is proportional to area (A).
Area is proportional to square of diameter(d)
hence, it is reasonable to consider P(d) = ad^2 + b
b) P(d) = ad^2 + b
In this formula, b= fixed cost
a d^2=cost of ingredients of pizza
c)P(d)= .04*d^2 + 4
P (10)= .04*10^2 + 4 =8
P (12)= .04*12^2 + 4=9.76
P (14)=.04*14^2 + 4=11.84
P (16)=.04*16^2 + 4=14.24
The cpomputed price are close to above price.
d) P (d) = .04*d^2 + 4
A = pi d^2/4
price per unit area = P/A = ( .04*d^2 + 4)/( pi d^2/4 )
=4(.04*d^2 + 4)/( pi d^2)
=(4/pi) ( 0.04 + 4/ d^2)
e) rate of change of price per unit area
d/dd ( dP/dA)
=(4/pi) (-4/d) <0
hence price per unit area is de- creasing as the size of the pizza increases.
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