A fence is to be built to enclose a rectangular area of 210 square feet. The fence along three sides is to be made of material that costs 4 dollars per foot, and the material for the fourth side costs 13 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.Dimensions
Question
A fence is to be built to enclose a rectangular area of 210 square feet. The fence along three sides is to be made of material that costs 4 dollars per foot, and the material for the fourth side costs 13 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.Dimensions
Solution
To solve this problem, we need to use calculus and the concept of optimization. Here are the steps:
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Let's denote the sides of the rectangle that are fenced with the 13 per foot material as y. The area of the rectangle is given by A = xy = 210 square feet.
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We can express y in terms of x using the area equation: y = 210/x.
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The total cost of the fence is given by C = 4x + 4x + 13y. Substituting y from step 2, we get C = 8x + 13(210/x).
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To find the dimensions that minimize the cost, we need to find the derivative of the cost function with respect to x, set it equal to zero, and solve for x. The derivative of C with respect to x is C' = 8 - (2730/x^2).
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Setting C' equal to zero gives 8 - (2730/x^2) = 0. Solving for x gives x = sqrt(2730/8) = 18.47 feet.
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Substituting x = 18.47 into the equation from step 2 gives y = 210/18.47 = 11.37 feet.
So, the most economical dimensions for the enclosure are approximately 18.47 feet by 11.37 feet.
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