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