Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Question

Answered step-by-step

Let $ a $ and $ b $ be positive numbers. Find the length of the shortest line segment that is cut off by the first quadrant and passes through the point $ (a, b) $.

Video Answer

Solved by verified expert

This problem has been solved!

Try Numerade free for 7 days

Like

Report

Official textbook answer

Video by Chris Trentman

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

04:52

Wen Zheng

01:23

Amrita Bhasin

Calculus 1 / AB

Calculus 2 / BC

Chapter 4

Applications of Differentiation

Section 7

Optimization Problems

Derivatives

Differentiation

Volume

University of Michigan - Ann Arbor

University of Nottingham

Boston College

Lectures

04:35

In mathematics, the volume of a solid object is the amount of three-dimensional space enclosed by the boundaries of the object. The volume of a solid of revolution (such as a sphere or cylinder) is calculated by multiplying the area of the base by the height of the solid.

06:14

A review is a form of evaluation, analysis, and judgment of a body of work, such as a book, movie, album, play, software application, video game, or scientific research. Reviews may be used to assess the value of a resource, or to provide a summary of the content of the resource, or to judge the importance of the resource.

11:37

Let $a$ and $b$ be positiv…

05:35

55, Let a and b be positiv…

01:49

Calculate the length and t…

02:51

00:38

Construct $\triangle \math…

00:34

01:39

Find the distance between …

were given to positive numbers A and B, and we have to find the length of the shortest line segment that is cut off by the first quadrant and passes through the point a. B Just get one. Well, first of all, equation of a line that passes through the point A B with a slope of em is why minus B equals M times X minus a. Now it's intuitively mm. Has to be less than zero. Otherwise, if we had a positive slope, there would not be a line segments cut off by the first quadrant. Now set X equal to zero. We can find the coordinates of the Y intercept. So we have a Y intercept with coordinates zero mhm negative AM plus B and an X intercept. You said Y equals zero, and we have that X is equal to negative B over M plus A. This is negative. Be over em, plus a zero. Yes. Therefore, the distance from the Y intercept to the X intercept shall call D as a function of the slope M. This is the square root of the difference of the X components, which is a minus B over M squared plus B minus AM squared. Now, of course, we know that she our function d is minimized at the same time when it's square is minimized, I'll call this function F. So F is a function of em is a minus B over em squared plus B minus A m squared. Now, to find the minimum of yes will differentiate and find where this is equal to zero. So f prime of em is two times a minus B over em times derivative of the inside. This is positive. Be over m squared plus two times B minus. I am in Pampers No times the derivative of the inside, which is negative A A. Now, if you simplify, we get to over M cubed times a B M minus B squared lawyers, plus a squared them to the fourth minus a b m cubed. We set this equal to zero. We can then factor by grouping so we get to over m cubed times and then we factor b out of the 1st and 2nd terms. We have B times a M minus B and we factor and a M cubed out of the second and 33rd and fourth terms. We get a minus. Yes, B sorry. A M minus b again, Just it is equals zero. I see it's going to be factored as to over m cubed times B plus A M cubed times a M minus B equals zero. And with this factory ization, well, as you've already pointed out, Okay, One possibility is that a M minus B equals zero. This would tell us that M equals B over a, which is positive. So this is not the answer. As we already pointed out, the slope needs to be negative. Therefore, the only solution is the other factor. B plus a m cubed equals zero. And so M is equal to so mhm negative cube root of B over a. Yeah. Now, in fact, a few think about the way we factored f prime of em. It follows that f prime of them. Mhm is because I think less than zero. If em is less than negative. Cuba to be over a and f prime of them is greater than zero. If m is greater than negative Cuba, it'd be over a. Therefore it follows that f has a minimum at M equals negative cube root of B over a. Yeah, I will plug this back into our distance function, so we have f of negative cube root of B over A. This is equal to once you finally simplify it down. There's a lot of steps here. I'm skipping. Mhm, a squared plus three a to the four thirds times B to the two thirds plus three. Eight the two thirds times B to the four thirds All right plus B squared. That's so I'm skipping some steps here, my peeps. But then the length is the square root of this. Now to simplify this, that we can actually factor this as yeah a to the two thirds plus B to the two thirds cubed, just huge me boy and therefore the distance a d of negative cube root of be over a is the square root of this, which is a to the two thirds plus B to the two thirds to the three halves. Power that. And so this is our shortest length

View More Answers From This Book

Find Another Textbook

Missouri State University

Campbell University

Idaho State University