26.07.2022 - 18:37

A landscape architect wished to enclose a rectangular garden on one side by a brick wall costing $30/ft and on the other three sides by a metal fence costing $10/ft. If the area of the garden is 42 s

Question:

A landscape architect wished to enclose a rectangular garden on one side by a brick wall costing $30/ft and on the other three sides by a metal fence costing $10/ft. If the area of the garden is 42 square feet, find the dimensions of the garden that minimize the cost.

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  • Shirley
    April 2, 2023 в 04:08
    Let the length of the rectangular garden be x and the width be y. The area of the garden is given by xy, which is 42 sq. ft. We need to minimize the cost of enclosing the garden, which is given by C = 30x + 10(2x + y) or simply C = 30x + 20y + 20x. Using the area equation, we can write y = 42/x. Substituting this into the cost equation, we get C = 30x + 20(42/x) + 20x = 50x + 840/x. To minimize the cost, we need to find the value of x that makes dC/dx = 0. Differentiating C with respect to x, we get dC/dx = 50 - 840/x^2. Setting this equal to 0, we get 50 = 840/x^2, which gives x = ?(840/50) = ?16.8 = 4.1 (rounded to one decimal place). Substituting this value of x into the area equation, we get y = 42/4.1 = 10.2 (rounded to one decimal place). Therefore, the dimensions of the garden that minimize the cost are \approx imately 4.1 ft by 10.2 ft.
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