How do you find the shape of a rectangle of maximum perimeter that can be inscribed in a circle of radius 5 cm?

Answer 1

Define the problem

Look at the figure below:

.

We will find the point #N(x,y)#
.

.
Perimeter, #P = 4x+4y#

In the question here, #r=5#, so we have:

#x^2 + y^2 = 25#, so

#y=sqrt(25-x^2)#

.

Therefore, the problem is:

Find #x# to maximize:

#P = 4x+4sqrt(25-x^2)#, with #0<= x <= 5#

Solution:
We find critical numbers:

#P' = 4 + (4x)/sqrt(25-x^2)#

#P'=0# at #x=-sqrt(25-x^2)#, so

#2x^2 = 25#, and so:

#x=5/sqrt2 = (5sqrt2) /2#

#P(0)=P(5)=20#

Whe #x = (5sqrt2) /2#, we also get #y = (5sqrt2) /2#, so

#P((5sqrt2) /2) = 4((5sqrt2) /2)+4((5sqrt2) /2) = 20sqrt2#

Because #20sqrt2 > 20#, the maximum value of perimeter occusu whan the rectangle is a square with sides: # (5sqrt2) /2# ,

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Answer 2

To find the shape of a rectangle of maximum perimeter inscribed in a circle of radius 5 cm, follow these steps:

  1. Recognize that the rectangle will be a square when it has the maximum perimeter.
  2. Determine the diameter of the circle, which is twice the radius, ( d = 2 \times 5 = 10 ) cm.
  3. The diagonal of the square (and also the diameter of the circle) will be ( d = 10 ) cm.
  4. Use the properties of a square to find the side length:
    • Since all sides of a square are equal, each side length will be half the length of the diagonal, ( \frac{d}{\sqrt{2}} = \frac{10}{\sqrt{2}} ) cm.
  5. Calculate the side length of the square:
    • ( \frac{10}{\sqrt{2}} ) cm ≈ 7.07 cm.
  6. The shape of the rectangle with maximum perimeter that can be inscribed in the circle of radius 5 cm is a square with side length approximately 7.07 cm.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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