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If you are offered one slice from a round pizza (in other words, a sector of a circle) and the slice must have a perimeter of $ 32 $ inches, what diameter pizza will reward you with the largest slice?

Diameter of pizza should be 16 inches

02:05

Wen Z.

01:08

Amrita B.

Calculus 1 / AB

Calculus 2 / BC

Chapter 4

Applications of Differentiation

Section 7

Optimization Problems

Derivatives

Differentiation

Volume

Campbell University

Harvey Mudd College

Idaho State University

Lectures

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for peace of around pizza. Okay, so there it is. If your piece must have a perimeter of 32 inches, what diameter pizza will reward you with the largest slice? Okay, Largest. So we're trying to maximize. Yeah. We don't want to say maximize the slice. Although that's what we're doing what it is. What is it about the slice that we're trying to maximize? We're trying to maximize the area of the slice maximize area of the slice. Yeah. Okay. And the constraint here is is that the perimeter of the slice has to be 30 two inches constraint. All right, so let's look at a slice of pizza here. Okay? So we're assuming that this is the center and that it's cut exactly straight. So these two things, or both are Okay, let's say the angle here is data. So they to find the area of sector of a circle, you need to know the sector of the circle formula. But if you're like me, I always forget it, so I always have to regenerate it. So here's what I do. Area of a circle fire squared over the angle to get all the way around. Two pi equals the area of the sector over the angle. Fada. Okay, so cross multiply across the pies out to get one half r squared data equals the area of the sector. So that's what we're trying to maximize. One half data are square. Okay, The perimeter of the slice, that would be R plus R. So our plus are plus this arc length. Here, try to find the arc length of ah, circle s equals R theta. So our theta has equal 32. So to r plus r theta equals 32. Let's go ahead and solve it for data. And then once we solve it for theater, we can plug it in right here. So r theta equals 30 to minus two Are that equals 32 minutes to our over our or 32 over R minus two. All right, so area is one half. 32 over. R minus, two times R squared. I should just left it the way it was, because now it's messy. One half 32 over. Our times are squared minus two overtimes R squared 16 R minus R squared. That's the area. Oh, okay. This one half canceled with the 32 left 16, and then r squared over. Our lives are on the top, and then this one have canceled with it to let me minus r squared. So I'm gonna take the derivative I get 16 minus two are and I said it equals zero. So two r equals 16. So r equals eight and wants the question I'm trying to find what diameter pizza will reward you with the largest slice. Well, the one whose radius is eight will be the one whose diameter is two times 8 16 inches. So if the pizza has dime or 16 inches, it will give you the largest slice who had, which has, Who has that has a perimeter of 32 inches.

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