3. (5 points) Normally, one expects the cost of the pizza per unit area to decre

Question

3. (5 points) 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 (b) How do the fixed costs appear in this formula? How does the cost of the ingredients of (c) Consider the function P(d) 04d2 + 4. Compute P(10), P(12), P(14), P(16) and P(d) = af + b where a and b are real numbers to be determined the pizza appear in this formula? compare with the prices above. (d) Use the formula P(d) 04d2 + 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

a) We know that C is the cost per unit area
And since area has two dimensions, it makes sense to use a quadratic to model this.

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b) P = ad^2 + b
where a,b are constants
P = price per unit area
and d = diameter of the pizza.

When d = 0,
P = b ---> fixed cost

So, the fixed cost heer is 'b' dollars

And the cost of ingredients is the 'a' dollars

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c)
P(10) = 0.04*10^2 + 4 ---> 4 + 4 --> 8

P(12) = 0.04*12^2 + 4 ---> 9.76

P(14) = 0.04*14^2 + 4 ---> 11.84

P(16) = 0.04*16^2 + 4 ---> 14.24

So, the values kinda match with the ones given in the question.

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d)
P = price
Area of the pizza = pi*d^2/4

Divide them we get the price per unit area thus :

C = (0.04d^2 + 4) / (pi*d^2/4)

C = 4(0.04d^2 + 4)/ (pi*d^2)

C = (0.16d^2 + 16) / (pi*d^2)

Another way to write it :
C = (0.16d^2/pid^2) + 16/(pi*d^2)

C = (0.16/pi) + 16/(pi*d^2) ---> ANS

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e)

Look at expression in 'd' above.

0.16/pi is a constant anyways.

When d increases, pi*d^2 increases
and since its in the denominator, 16/(pi*d^2) decreases
thereby showing this

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