Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

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) $.

shortest length is $\sqrt{a^{2}+3 a^{4 / 3} b^{2 / 3}+3 a^{2 / 3} b^{4 / 3}+b^{2}}=\left(a^{2 / 3}+b^{2 / 3}\right)^{3 / 2}$

01:23

Amrita B.

08:16

Chris T.

Calculus 1 / AB

Calculus 2 / BC

Chapter 4

Applications of Differentiation

Section 7

Optimization Problems

Derivatives

Differentiation

Volume

Campbell University

Harvey Mudd College

Boston College

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

01:39

Find the distance between …

06:30

Let $(a, b)$ be a fixed po…

00:57

Draw a segment $\overline{…

0:00

Let $ C $ be the point on …

Let $a$ and $b$ be positiv…

01:19

For each pair of points fi…

01:15

Consider the segment of th…

02:39

02:14

Find the length of the lin…

01:49

Determining a line Segment…

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

Numerade Educator

In mathematics, integration is one of the two main operations in calculus, w…

In grammar, determiners are a class of words that are used in front of nouns…

Find the distance between the two points and the midpoint of the segment joi…

Let $(a, b)$ be a fixed point in the first quadrant and let $S(d)$ be the su…

Draw a segment $\overline{A B}$ . Construct a segment $\overline{X Y}$ whose…

Let $ C $ be the point on the line segment $ AB $ that is twice as far from …

Let $a$ and $b$ be positive numbers such that $a<b$. In which quadrant is…

For each pair of points find the distance between them and the midpoint of t…

Consider the segment of the line $y=m x+c$ on the interval $[a, b] .$ Use th…

Find the distance between the pair of points and find the midpoint of the se…

Find the length of the line given by $\mathbf{r}(t)=\langle t, 2 t\rangle,$ …

Determining a line Segment with Given Mid- point Let $(4,4)$ be the midpoint…

02:33

Show that $ \displaystyle \int_0^\infty e^{-x^2}\ dx = \displaystyle \int_0^…

01:25

Astronomers use a technique called stellar stereography to determine the den…

06:13

Evaluate the integral.

$ \displaystyle \int_0^1 x \sqrt{2 - \sqrt{1 …

08:57

Find the values of $ p $ for which the integral converges and evaluate the i…

04:07

For each initial approximation, determine graphically what happens if Newton…

07:41

What is the shortest possible length of the line segment that is cut off by …

02:21

Find the escape velocity $ v_0 $ that is needed to propel a rocket of mass $…

00:41

Use Newton's method with initial approximation $ x_1 = 1 $ to find $ x_…

02:05

Use a computer algebra system to evaluate the integral. Compare the answer w…

02:42

Find the dimensions of a rectangle with area $ 1000 m^2 $ whose perimeter is…

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.