SOLUTION: Suppose you want to enclose a rectangular backyard that is adjacent to the ocean. The backyard area must contain 12000 sq ft. No fencing along the ocean. What dimensions will use t
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Question 1000878: Suppose you want to enclose a rectangular backyard that is adjacent to the ocean. The backyard area must contain 12000 sq ft. No fencing along the ocean. What dimensions will use the least amount of fencing? Answer by Theo(13342) (Show Source):
assuming the length is parallel to the ocean, only 1L is required because the ocean side will be open.
the perimeter is therefore equal to L + 2W.
from the equation of L * W = 12000, you can solve for L to get L = 12000/W.
in the equation for perimeter, replace L with 12000/2W to get p = 12000/W + 2W
if you have a graphing calculator, you can solve this graphically.
if you don't, you can solve this using calculus by finding the derivative of the function and then setting that to 0.
either way you will see that the minimum point on the graph is when x = 77.459661 and y = 309.83867.
this means that the perimeter is equal to 309.83867 when the width is equal to 77.459661.
since the area is equal to length * width and the area is 12000 and the width is 77.459661, you can solve for length to get length = 154.9193457
when width = 77.459661 and length = 154.9193457, the perimeter is equal to L + 2W which is equal to 309.83867 and the area is equal to L * W = 12000.
the graph of the equation is shown below.
this graph was not good to find the value because the results are rounded too much.
i used the TI-84 plus to graph the equation and find the minimum point.
only the side of the graph where x >= 0 was used since negative values for x or y are not allowed.
to solve using calculus, you find the derivative.
the derivative of 12000/w + 2w^2 is equal to -12000/w^2 + 2.
set it equal to 0 and you get -12000/w^2 + 2 = 0
subtract 2 from both sides to get -12000/w^2 = -2
divide both sides by -12000 to get 1/w^2 = 2/12000 = 1/6000
this equation is true when w^2 = 6000
take the square root of 6000 to get w = 77.45966692.
since area = LW and W = 77.45966692 and area = 12000, you can solve for L to get L = 12000 / 77.45966692 = 154.9193338.
area = 77.45966692 * 154.9193338 = 12000
perimeter = 2 * 77.45966692 + 154.9193338 = 309.8386677.
your perimeter is 309.84 rounded to 2 decimal places.
your width is 77.46 rounded to 2 decimal places.
your length is 154.92 rounded to 2 decimal places.
any discrepancies in the numbers used for width and the length and the perimeter are due to rounding.
different calculators round to different numbers of decimal places.
the graphing calculator also shows results rounded to different decimal places.
the end result is the same when you round to 2 decimal places because the differences aren't showing up until around the 5th decimal place.
