Problem 39 Hard Difficulty

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?

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Video Transcript

suppose you want to find the diameter of the largest slice of pizza With the perimeter of 32". Now this slice of pizza is just a sector of this circle and so consider the illustration below. To begin, we first find our objective function. Note that the objective function is the one being maximized in this problem. Since you want a larger slice then the area of the sector must be maximized. Thus the objective function instead of the area of the sector which is a that is equal to Data over 360 times pi R squared. Next you want to write this objective function in terms of one variable only to do this. We used to give an information that the Perimeter of the slice must be 32". Now this perimeter is equal to the some of the radius plus the arc length. That is p this is to R plus our data and f b h 32 we have 32 equal to two R plus our data from here. We want to solve fourth data in terms of our so we have our data, this is equal to 30 to -2. Are or Data is just 32 over AR -2. Substituting this to our objective function. We have a which is equal to 32 over AR -2 over 3 60 times by R squared. Or this is just By over 360 Times We have 32 over ar minus two times R squared or let's just buy over 3 60 times 32 AR -2 R Squared. Now that we have written our objective function in terms of one variable. We are now ready to use concept of relative extreme. 1st. You want to find the derivative of a So if a a spy over 360 times 32 ar minus two R squared, then a prime is just By over 360 times 32 minus for our. And then from here we want to set a prime zero. And so for our and we have Pi over 360 times 30 to minus four. Art is equal to zero. And so we get 30 to minus four are equals zero or Negative for our that's equal to negative 32 or that are is trust equal to eight. To check of this radius maximizes a. We apply 2nd derivative test. You want to find the second derivative of a that is A double prime which is equal to pi over 360 times you have negative for and so we have Negative Pi over 90 which is always negative for any value of or and by second delivery test this means that the Area is maximized when R is equal to eight and so far the largest slice The radios must be eight or that the diameter must be D which is twice the radius or that is just two times eight Which is 16