a landscape engineer has 200 ft of border to enclose a rectangular pond. what dimensions will result in the largest pond? please show clear steps. thank you in advance, i hope to learn from your answers.
2007-10-26 18:31:49 · 3 answers · asked by Jake K 1 in Education & Reference ➔ Homework Help
A square is the largest volume per perimeter. So 200 = 4x
x = 50; Make your dimensions 50 x 50.
2007-10-26 18:52:29 · answer #1 · answered by Anonymous · 0⤊ 0⤋
Tinger got it nearly right. I'd use "area" not "volume," but he's right about the square and the largest area in a two-dimensional world.
FWIW- the greater the number of equilateral sides, the larger the area you can enclose with a perimeter. If you allow a circle to have an infinite number of sides, you can get the most area from your material if you enclose a circle. But that contradicts the premise of your question, which cites a "rectangle." I believe a square is a rectangle, so I'll go with Tinger's dimensions.
2007-10-27 02:10:53 · answer #2 · answered by going_for_baroque 7 · 0⤊ 0⤋
The problem is to find the rectangle with maximum area for a given perimeter.
If P is the perimeter, l the length, and w the width:
w = 0.5P - l ........................(1)
area a = l x w = (0.5P - l) x l = 0.5P x l - l^2
For a to be maximum the derivative da/dl = 0.
da/dl = 0.5P - 2l = 0
l = 0.5P/2 = 0.25P
From (1): w = 0.5P - 0.25P = 0.25P = l
The rectangle has to be a square.
Conclusion: For a given perimeter a square has the greatest area.
For this problem P = 200 ft.
The pond size is a square of side = 0.25P = 50 ft.
2007-10-27 02:19:33 · answer #3 · answered by A.V.R. 7 · 0⤊ 0⤋