Find the dimensions of the rectangle of maximum area whose perimeter is 16 cm ?

Answer 1

The rectangle of maximum area is a square of side length #l=4# cm

Let #x# and #y# be the lengths of the sides of the rectangle measured in cm.

Then the perimeter is:

#2x+2y = 16#

so that:

#y = 8-x#

The area is then:

#S = x*y = x(8-x) = 8x-x^2#

Find the critical points of the function:

#(dS)/dx = 0#
#8-2x = 0#
#x=4#

and as:

#(d^2S)/dx^2 = -2 < 0#

the critical points is a maximum.

Then the maximum area is obtained when #x=4# and #y=8-x=4#, that is when the rectangle is a square.
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Answer 2

Let the length of the rectangle be ( l ) cm and the width be ( w ) cm. Given that the perimeter is 16 cm, we have the equation:

[ 2l + 2w = 16 ]

Solving this equation for ( l ), we get:

[ l = 8 - w ]

The area of the rectangle, ( A ), is given by:

[ A = lw ]

Substituting ( l = 8 - w ) into the area formula, we get:

[ A = (8 - w)w ]

To find the maximum area, we can take the derivative of ( A ) with respect to ( w ), set it equal to zero, and solve for ( w ):

[ \frac{dA}{dw} = 8 - 2w ] [ 8 - 2w = 0 ] [ w = 4 ]

Substituting ( w = 4 ) back into the equation for ( l ), we find:

[ l = 8 - 4 = 4 ]

Therefore, the dimensions of the rectangle of maximum area with a perimeter of 16 cm are ( 4 ) cm by ( 4 ) 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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