Optimization: Find Dimensions of Rectangle With Largest Area Inscribed in a Circle

This video shows how to find the dimensions of a rectangle with largest area that can be inscribed in a circle of radius r.

[@cesium8857]

your accent helps me pay attention i have no idea why

[@ethandoest9426]

This video is going to save my final grade!! This solution makes sense now!!

[@crowsmathclass]

Glad it helped. Good luck in your class.

[@幻彩小羽毛]

I have a homework question with the exact same same question lol, this was really helpful

[@gartyqam]

calculus early transcendentals section 8th edition 4.7 question #25

[@Maya-xs9xn]

@@gartyqam It's a very common Calculus book for college.

[@kainime7361]

Sir I am new to your channel and your explanation amazing it is clear as a crystal of water

[@oliviawoloshyn5824]

Love your voice and this explanation is really easy to follow big thumbs up

[@crowsmathclass]

Thank you. Glad it helped.

[@Waltu0]

Shouldn't the second term for the derivative of the area be -4x^2/sqrt(r^2-x^2) instead of -4x/sqrt(r^2-x^2)? -x*4x= -4x^2.

[@Kakashi713]

He did input it there unless I'm missing something? His answer is right.

[@rezanhammo3224]

Midterm exam tomorrow, I really appreciate this video

[@crowsmathclass]

Good luck on your exam.

[@nam3go3sh3r3]

Very helpful, thank you!

[@crowsmathclass]

You’re welcome. Glad it helped.

[@bas5rocker311]

Thankyou sir!. This was pretty helpful.

[@crowsmathclass]

You’re welcome. Glad it helped.

[@lafhee]

I LOVE YOUR ACCENT

[@crowsmathclass]

Thank you.

[@Maya-xs9xn]

Thank you, it really help me understand the problem.

[@crowsmathclass]

You’re welcome. Glad it helped.

[@adamvaught6611]

How did the r disappear at the end? Shouldn’t it be r-r/2^1/2?

[@ramimazaoui]

it's like 1-1/2 = 1/2

[@jfig7682]

so are we solving for 2x at the end rather than singular x or y?? i was bit confused.

[@crowsmathclass]

x

[@laurensalino]

Super super helpful

[@crowsmathclass]

Glad it helped.

[@eduardobarragan1768]

How would you find the radius of the entire circle?

[@Somm_RJ]

The radius is a variable. Hypothetically, you can plug any number to it.

[@RahooMohammadi-cz8es]

Can you please help me So i was thinking what if the tye rectangle has a length of 11 cm and width of 6 cm. How do we solve that ??

[@Fingerlickinggoood]

just input the numbers into x and y I think

[@maheralabbar189]

I have a question. Why can’t the area just be xy. Like would it still work. I tried it with radius 4 and I am getting 8 as my area. I don’t get why it has to be (2x) (2y) is there something else to do instead of splitting it into fourths

[@crowsmathclass]

Because center of circle is at the origin.

[@maheralabbar189]

@@crowsmathclass so is it necessary to split it up. My teacher taught us to have the area as xy

[@crowsmathclass]

Yes. Because we center at origin.

[@maheralabbar189]

@@crowsmathclass is there a situation where we can just use A= xy without 2x 2y Like how would we know if it is centered and not centered

[@invinceble5882]

i love your accent

[@crowsmathclass]

Thank you.

[@faithfury2397]

Hi! i have a question, why r^2 is a constant for this example? i saw time rates with the radius getting derived at, isn't it the same with optimization?

[@crowsmathclass]

We are looking for the dimensions of the rectangle. r is a constant in this problem. So we are looking for x and y.

[@Somm_RJ]

Why would you need to do all that? A square is a rectangle. You just have to draw a line (radius) from the center, let's call this A. Then, another line (radius) at 90° of the first line, Let's call this B. Now, you already have 2 sides of a triangle, just calculate the hypothenus using Pythagorean Theorem using the 2 sides above, sqrt(A^2+B^2). Because the 2 sides (A and B) are equal, same as the length of the radius (r), you can simplify this by doing sqrt(2r^2). Or you can alternatively, you can use the sine function, r/sin 45°, since the resulting triangle would be an isosceles triangle and you have a given 90° on 1 angle. The result would be 1 side of the rectangle. Since it is a square, then you just multiply it by itself. In simplest form, it would look like this, Area=sqrt(2r^2)^2 Alternatively, Area=(r/sin 45°)^2

[@qalbabbas2496]

You should find second derivative and find Local maximum and minimum for area,which can give surety for maximum area

[@natalie-rg9xm]

Thanks man. nice voice

[@crowsmathclass]

You’re welcome. Thank you for the comment.

[@qalbabbas2496]

How can you say that area is maximum

[@beefylilun178]

As proof, use the 1st or 2nd derivative test. I think it will show you that that is indeed the maximum area

[@francescovellani2057]

Thank u so much

[@crowsmathclass]

You're welcome. Glad it helped.

[@deadlysummer8071]

Thank you sir

[@crowsmathclass]

You’re welcome.

[@pmat2757]

My god thank u so much

[@crowsmathclass]

You’re welcome.

[@victorazariah9038]

👍

[@Anton-lv9fw]

u should go faster

[@kainime7361]

Sir I am new to your channel and your explanation amazing it is clear as a crystal of water

[@crowsmathclass]

Thank you for the nice comment.

[@tashinikaaponsu4]

Thank you sir

[@crowsmathclass]

You’re welcome.