find the dimension of a rectangle with perimeter of 100 m and the largest area, calculus 1 tutorial

find the dimension of a rectangle with a perimeter of 100 m and the largest area
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[@gammano0b858]

I solved by using the fact, that if you have a rectangle with fixed perimeter, and you want it's area to be maximized, the rectangle will be actually a square, so perimeter = 100 => square side length = 25 => area = 625.

[@ok-ez1xl]

i did in a similar way...
after calc the eqn of the dimensions 50x-x² = max
So I applied max value of a quadratic i.e. ofc +ve in the case( I was saved)
So max value = -D/4a = 625.
So 50x-x²=625
So, x=25 , 50-x = 25.

[@johnnolen8338]

Also it can be proved as a theorem that a rectangle of maximum area is a square (using calculus) 😎

[@popodori]

if x=50, then y=0 and the area = 0, vice versa, if y=50 then x=0. (assuming distances have to be postive) so for 00 the area is positive, thus fair to assume the maximum is at x=y, ie 4x=100, x=y=25

[@johnnolen8338]

@@popodori You don't have to assume anything in order to show that a rectangle enclosing a maximum area is a square.
Start with a fixed perimeter: P = 2(x + y). For this case, y = P/2 - x. The enclosed area is given by A = xy = x(P/2 - x). By the product rule, A'(x) = - x + (P/2 - x) = 0. Therefore P/2 = 2x => x = P/4. Since y = P/2 - x, y = P/2 - P/4 = P/4. A'(x) = P/2 - 2x => A"(x) = -2. Since A" < 0, A is a maximum at the critical point (P/4, P/4). Let x = y = P/4 = s then P = 4s, and A = s^2. *Conclusion* : A rectangle of maximum area is always a square.
In this specific problem the perimeter, P = 100. That means 100 = 4s. Thus s = 25. ◼

[@oom_boudewijns6920]

@@johnnolen8338 but a square is not identified as a rectangle right

[@user-dl8nk5bf8v]

4xy = ( x + y )^2 - ( x - y )^2

[@romeisbig6485]

Sorry if it sounds stupid...
How do you find the minimum area possible ?

[@arnavkulkarni2370]

If we want the minimum area, the length would be infinite and the breadth would be just near to 0
Which will give a straight line whose area is nearly 0
You're Welcome😉

[@romeisbig6485]

@@arnavkulkarni2370 Thanks
I thought so too
But, is there a way to come to this conclusion using derivatives?

[@michaelraad857]

@@romeisbig6485 Hey, I'm a bit late but:
The equation for the area in this video was A=-x^2+50x. Factor and you get A=-x(x-50)
We want to know when the area is zero (the minimum area as it cannot be negative) so we set the area equation to zero. We learn that when x=0 or x=50, the area reaches 0. The length is not infinate as the rectangle must have a perimeter of 100, however one side approaching 50 as the other approaches 0 would thin the rectangle down almost to a line (saying the dimentions are 50 and 0 would be wrong as it would be an actual line rather than a rectangle)
No need for derivatives :p

[@littleretroship6403]

I love your content sir! You have greatly helped me with learning differential calculus!
Edit: I have a guess: 225m^2
Both sides 25m
I remember noticing that for rectangles, squares have the biggest area with the same perimeter. But also optimisation if I didn’t know that

[@blakedylan8387]

I was so ready for you to get those Lagrange multipliers out since we were dealing with two variables!! I forgot you can do this without multivariable calculus 😂😂

[@chessthejameswei]

We really grinding for that AP Calc Perfect score aren't we now 👍

[@exynosnemea2937]

We can actually use this method to find the maximum possible volume of a box.

[@rafaelgcpp]

Why don't you evaluate the second derivative on the critical number? If it's positive, then is a minimum, if it's negative, it is a maximum, and if it's zero, it is a saddle point

[@jqrn]

I got confused by the question. Find a rectangle... To me it would seem that with 25 side length, it's a square, not a rectangle. I thought i knew the difference of the two, but now I know I will never understand the difference between square and rectangle

[@bozzisimone]

The square is a particular rectangle where all the sides have the same length

[@user-pm6rl1rp4g]

But u can use the Cauchy inequality…

[@coderanger7708]

What will happen if the question says that the rectangle has the minimum area?

[@TalSzor]

infinitely thin and long rectangle

[@coderanger7708]

@@TalSzor but if it's infinitely long means the perimeter will not be 100 is it, so what about a line of length 100 units

[@TalSzor]

@@coderanger7708 right, sorry. only infinitely thin, but finitely long (50m long to be precise)

[@coderanger7708]

@@TalSzor yeah, thanks

[@arnavkulkarni2370]

USE AM-GM MAN

[@teelo12000]

But then you reach that great philosophical debate: does a square count as a "Rectangle"?

[@granieiprogramowanie2235]

by definition.. YES

[@hrayz]

@@granieiprogramowanie2235 , yup. By definition a square is just a specific rectangle, also a specific trapezoid.

[@kunjukunjunil1481]

All squares are rectangles but all rectangles are not squares