Using the side of a building as one side and fencing for the other three sides, a rectangular garden will be constructed. Given that there are 60 feet of fencing available, determine the dimensions that would create the garden of maximum area and identify the maximum area. Enter only the maximum area. Do not include units in your answer.

Question

Mathematics

Sullivan Hensley

11 December, 00:00

Using the side of a building as one side and fencing for the other three sides, a rectangular garden will be constructed. Given that there are 60 feet of fencing available, determine the dimensions that would create the garden of maximum area and identify the maximum area. Enter only the maximum area. Do not include units in your answer.

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Answers (1)

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Serena Francis · Answer 1

The dimensions that would create the garden of maximum area are 30 feet and 15 feet

The maximum area is 450 feet²

Step-by-step explanation:

The garden is fencing for three sides only, That means the length of the fencing is equal to the sum of the length of the three sides

Assume that garden is y feet long and x feet wide and the fencing will cover on side of y feet and two sides of x feet

∵ The sum of the length of the 3 sides = y + x + x

∴ The sum of the length of the 3 sides = y + 2x

- The length of the fencing is equal to the sum of the sides

∵ The fencing is 60 feet

- Equate y + 2x by 60

∴ y + 2x = 60

- Find y in terms of x by subtracting 2x from both sides

∴ y = 60 - 2x

To find the dimensions which make the maximum area, find the area of the garden, then substitute y by x, and differentiate it with respect to x, then equate the differentiation by 0 to find the value of x, and substitute this value in the equation of y to find y and in the equation of the area to find the maximum area

∵ The formula of the area of a rectangle is A = l * w

∵ l = y and w = x

∴ A = x y

- Substitute the value of y above in A

∵ A = x (60 - 2x)

- Multiply bracket by x

∴ A = 60x - 2x²

Now differentiate x with respect to x

∵ A' = 60 (1) - 2 (2) x

∴ A' = 60 - 4x

- Equate A' by 0 to find x

∴ 0 = 60 - 4x

- Add 4x to both sides

∴ 4x = 60

- Divide both sides by 4

∴ x = 15

- Substitute the value of x in the equation of y to find it

∵ y = 60 - 2 (15)

∴ y = 30

The dimensions that would create the garden of maximum area are 30 feet and 15 feet

To find the maximum area substitute x by 15 in the equation of the area

∵ A = 60 (15) - 2 (15) ²

∴ A = 900 - 450

∴ A = 450

The maximum area is 450 feet²