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A Norman window is one in which the window is constructed by capping a rectangular region with a semicircular region (see Figure 11 ). If the total perimeter of the window is to be 30 feet, find the dimensions that maximizesits area.

radius $=$ height of rectangle, $x=\frac{30}{\pi+4}, y=\frac{60}{\pi+4}$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 4

Applications I - Geometric Optimization Problems

Derivatives

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

06:26

A Norman window is constru…

01:19

A Norman window consists o…

04:16

A Norman window has the sh…

05:10

04:12

A window is in the form of…

10:52

Maximum Area A Norman wind…

um and these two problems we have, we want to we have a a fence system offense or plot what we want to actually draw a little picture here for you. You can look at a book but like this and then halfway across we have this and then halfway across that we have this. So this is why this is why over to this is X over two and this is X. So we want the area, the total area of all of this to be 480 m2. We want to minimize the perimeter. So the perimeter we have, you have two X plus another X. So we're on over three X. Then we have two Y plus half of a half A Y. And that equals um so we get five halves Y. Actually we didn't really even need that. This thing could be any word here, right? Doesn't matter have the same problem. So this for why substituted in here and we get the perimeter is three X plus um 1200 over x. Take the derivative, set it equal to zero. We get three minus 1200 over x. One squared equals zero. That tells us that X one equals 20 m. And if we substitute that back into here, you get why one of course 24 m. So we would have this would be 24 this would be 10, 20 should be 20 and that would minimize the amount of fence that we need to get this area. Now we're given this thing called norman window which um let me just draw maybe I can do a quick little sketch here in a circle and cut off half of that circle. Maybe too much of that. But anyway, and then we have um a square underneath here. Yeah, something like that. Um So this is a circle And it's a it's a complete half circle here. And then um this is just a square. So we want uh we want the total perimeter of the window to be 30 ft and we want to maximize the find the dimensions that maximize the area. So this was this was why? So the perimeter, the perimeter of the circle here, is there half the circumference or or um hi are, which is and our is our why over two. So we get high y over two. Then we have this piece here. So you have to add Y. And then we have two X. So our circumference is one plus pi over two times Y Plus two Acts equals 30. Now the area is the area of this and the in the area of this. So the area of the rectangle is X times Y. In the area of this 1/2 circle is one half times um Hi I did I did most of this one off. Let's do it in real time. Then I forgot a pie in here. So we need to put pie into their and so then we need let's see here that we look back at my notes for a little bit. Uh Did I did I put the pie in my notes. No I did not. So let's see here um we have high there. So we have um that should be uh yeah and that this should be um wait a minute right here. This should be much simpler than that. What did I do know? Um Oh I know that. We need to, let's see here. Simplify that down a little bit and we get let's see here minus uh minus 18. Why don't want that minus 1/8 Y. And then we get minus 1 20 plus four plus plus five times Y. So this would be plus four. This pie plus four four span All right. And so now this is obviously wrong, we can take the derivative and set that equal to zero. And that gives us um the equation. I'll see here. We have um 15, 15 minus why over four times why four plus pi And that tells us that why one? This will be why one a critical point. So we can get rid of all this. Get rid of this rules. Rules that all? That's not right. So we'll fix it up. So why the critical point critical value is where Y equals um 60/4, 4 plus pi 60. All over four plus five, which numerically is uh 8.4 Big .4. And these were in feet. And so if that is let's see what X is. Um and X equals um 30/4 pi four plus pi say. And numerically that is 4.2. Of course, half of that. That makes some sense. And so then the area is uh let's see here, calculate that. And so the area was up being 63 square feet. All right. So that makes a little more sense if we have a pie in there, I wish I was short changing the area of the little top part here because I didn't multiply by pi. All right. I think that's good.

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