How do you find the dimensions of the rectangle of greatest area whose perimeter is 20 cm?

Answer 1

#5 "cm" xx 5 "cm"#

Let the length of one side of the rectangle be #x# #"cm"#.
Then the opposite side is also of length #x# #"cm"#, while the two other sides are of length:
#(20-2x)/2 = 10-x# #"cm"#

The area of the rectangle is:

#x(10-x) = 10x-x^2 = 25-25+10x-x^2 = 25-(x-5)^2# #"cm"^2#
This attains its maximum, #25#, when #x=5#
Hence the rectangle of maximum area is a #5 "cm" xx 5 "cm"# square.
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Answer 2

To find the dimensions of the rectangle of greatest area given a fixed perimeter, you can use the method of calculus optimization. Let ( l ) be the length of the rectangle and ( w ) be the width.

  1. Write down the formula for the area of the rectangle: ( A = lw ).
  2. Write down the formula for the perimeter of the rectangle: ( P = 2l + 2w ).
  3. Use the given perimeter to express one of the variables in terms of the other. Here, we can express ( l ) in terms of ( w ) as ( l = \frac{20}{2} - w = 10 - w ).
  4. Substitute the expression for ( l ) in the area formula: ( A = (10 - w)w ).
  5. Expand the equation: ( A = 10w - w^2 ).
  6. To maximize the area, take the derivative of ( A ) with respect to ( w ) and set it equal to zero: ( \frac{dA}{dw} = 10 - 2w = 0 ).
  7. Solve for ( w ): ( 10 - 2w = 0 \Rightarrow w = 5 ).
  8. Substitute the value of ( w ) back into the expression for ( l ): ( l = 10 - 5 = 5 ).
  9. Therefore, the dimensions of the rectangle of greatest area are ( l = 5 ) cm (length) and ( w = 5 ) cm (width).
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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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