💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Watch this step-by-step video, matched to your homework problem.

Try Numerade Free for 30 Days

Like

Report

In the following exercises, solve the given maximum and minimum problems.What are the dimensions of the largest rectangular piece that can be cut from a semicircular metal sheet of diameter $14.0 \mathrm{cm} ?$

Calculus 1 / AB

Chapter 24

Applications of the Derivative

Section 7

Applied Maximum and Minimum Problems

Derivatives

Campbell University

University of Nottingham

Idaho State University

Boston College

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

08:48

Maximum Area Find the dime…

06:54

For the following exercise…

03:14

01:20

04:20

01:42

07:51

A rectangle is constructed…

05:41

Rectangles beneath a semic…

05:27

Find the dimensions of the…

04:32

the problems that extra 25. And in that problem, um, we are drawing a rectangle and that they might revoke have a radius of like this and number 25 the equation of the you that y equals route 25 minutes X squared. That's a circle with a radius of by. So let's just make this cur semicircle to have a radio are the equation will be r squared, r squared minus X squared. I mean, you need point on the circle has courting a X comma route North Korea minus an actuary. All right, now to draw are tingling type of circle we can draw out like this. Take a point on the circle and, um, jaw, the horizontal and vertical line and the dimension of the circle we can feed at the base. Ah, there is an ex because at this point, this difference of acts and the semicircle in a cemetery This is negative. Negative X You didn't left or you're just thinking from the positive defeated Malcolm X and the height or around it a width of if you can go. Is that why, according which is why which is rude are three minus x squared. For the area of this rectangle be written a function of X, the T Rex. Absolutely conduit, which is our quarter mile X squared, the vertical distant care. All right, And now, to we just need to find a minimum of the function. Uh, someone start by taking the derivative, you find a critical number. So it kind of X, we'll find out another page. You're finding the derivative we need to use first about it. Product rule, because you're mortifying to function. You come in are two X and for my next great. So first we have the derivative of I would suggest to her, too. I'm the other function our quarter mile square and then plus, keep the first forms of the same differentiate the second function you do that, we have to use the change roles while the parable. Yeah, I know. The square root means power of 1/2 even a power. All of that 1/2 times a power of negative 1/2. Which means that the square root in the denominator one over to root. Are you in my quarters? Remember that are the constant. And then the general said we have to multiply this by the derivative of the inside. The driven of that bar scored the constant that 10 You were just doing the negative X Square has become. Do you ever know that of negative? So here, the derivative according to the project, that symbol, it is a little bit so, huh, Joe? Our story, my friend. What, uh, give me It will be a Maya since my time r minus this to natural. Can't go out and there still be a two at the numerator. So we're going to have to Don't you actually make x squared over? Roots are squared minus x. All right. And to find the critical numbers we need to set the derivative you Court of Europe. And we, uh, we can also check worth under. Fine. Um, this derivative will be undefined. You're under final X equals r. That word make a division by zero happen here. But if x equals r thing is this distance extend all the way to the edge of a struggle. So Mmm. Um, the point is actually all the way over here, the edge of the struggle. In that case, we don't get a rectangle at all. Or you could think you would've directing with area of a hero. And we're trying to look for a rectangle maxim area. So we're not even the mexica with art is the critical number. We're not going to, uh I'm not going to be concerned with that. I could really go for the maximum area rectangle. All right, Proper X In this equation, when the derivative is equal to zero, we can add a terminal outside. I'm gonna make it positive to expert over route are squaring my sex. Great. You go. I have to wrote, um, are recorded in effect for it, right? Like a motive. Quite a square root term hearing the denominator to both sides of the equation. When you do that on the right side, that's going to combine with corporate here already and according to cancel out so on the outside. But it's going to be left to X squared. No, I think we'll have to time this part. Uh, without, of course. So if they're going to be in quantities are square tomorrow. All right. Some of my things at the buy side by two. That'll give a X squared equals just r squared minus X square. Now move the export of death by giving a to X squared equals r squared. I'm taking X squared equals 1/2 arc squared and we discovered that we get plucked a minus. Um, uh, wonder So am I. Everyone have art, which we modified or rationalize your two over too are. And again, let's go back to their version of problems. The if X is negative. That just means at this point that we've chosen is actually to the left breath corner correspond to the same the same kind of rectangle, the negative activity here, the other carbon rectangle connected. It went over here, so we don't need to worry about the positive route to what you are. We're going to use that. And so this is the only critical number we're concerned with, so it must be the It's the only critical number. It must be the place where the area is maximum maximized. I and remember that problem? We're asking for the dimensions of the rectangle. About the dimension of the rectangle are they are the base and with basic to X. Oh, um, base. So, um, actually just found with Rachel. What you are. But I mean that the dimension of the fish or the base They've just right. You are other idiot, whatever the radio. And then for the day I'm pretty other side of the rectangle. That's why why we can use this equation y perfect route are escorting my next word with X square where we caught it early Here it's 1/2 r squared 1/2 bar squared. Uh, yeah, What happened? R squared. So tracking those to give one happen squared inside of crude And again this will rationalize to two But what you are the dimension of the rectangle inside the semicircle with Maxim area is when the X is rich over to our You didn't hire sigh here. You know what you are? Uh, Rick, you're and the this out of the rectangle. Yeah. Richard, What? You are

View More Answers From This Book

Find Another Textbook

In mathematics, precalculus is the study of functions (as opposed to calculu…

In mathematics, a function (or map) f from a set X to a set Y is a rule whic…

Maximum Area Find the dimensions of the largest rectangle that can be inscri…

For the following exercises, consider a wire 4 ft long cut into two pieces. …

For the following exercises, set up, but do not evaluate, each optimization …

For the following exercises, draw the given optimization problem and solve.<…

For the following exercises, set up and evaluate each optimization problem.<…

A rectangle is constructed with its base on the diameter of a semicircle wit…

Rectangles beneath a semicircle A rectangle is constructed with its base on …

Find the dimensions of the rectangle with maximum areathat can be inscri…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.