How do you find the length and width of a rectangle whose area is 900 square meters and whose perimeter is a minimum?

Answer 1

Length = Width = 30 meters

If #LxxW = 900# #rArr L= 900/W#
#P = 2*(L+W)#
#color(white)("XX")##= 2(900/W +W)#
#color(white)("XX")##=1800W^(-1) + 2W#
The minimum will occur when #(dP)/(dW) =0#
#(dP)/(dW) = -1800W^(-2) +2 = 0#
#color(white)("XXXX")#1/W^2 = (-2)/(-1800)#
#color(white)("XXXX")##W^2 = 900#
#color(white)("XXXX")##W =30#
and since #LxxW =900# #rArr L =30#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the length and width of a rectangle with an area of 900 square meters and a minimum perimeter, you can use calculus.

Let ( l ) be the length and ( w ) be the width of the rectangle.

  1. Write the formula for the area of a rectangle: ( A = lw ).
  2. Write the formula for the perimeter of a rectangle: ( P = 2l + 2w ).
  3. Since ( A = lw = 900 ), express ( l ) in terms of ( w ): ( l = \frac{900}{w} ).
  4. Substitute ( l ) into the perimeter formula to get it in terms of ( w ): ( P(w) = 2\left(\frac{900}{w}\right) + 2w ).
  5. Simplify the equation: ( P(w) = \frac{1800}{w} + 2w ).
  6. To find the minimum perimeter, differentiate ( P(w) ) with respect to ( w ) and set it equal to zero: ( \frac{dP}{dw} = 0 ).
  7. Solve for ( w ) to find the critical points.
  8. Determine if the critical points correspond to a minimum perimeter.
  9. Once you find the value of ( w ), use it to find the corresponding value of ( l ) using the equation ( l = \frac{900}{w} ).
  10. These values of ( l ) and ( w ) will give you the dimensions of the rectangle with the minimum perimeter.
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7