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## A family wants to build a rectangular garden on one side of a barn. If 600 feet of fencing is available to use, then what is the area of t

Question

A family wants to build a rectangular garden on one side of a barn. If 600 feet of fencing is available to use, then what is the area of the largest garden that could be built?

(A) Define a function that relates the area enclosed by the fence (in 〖ft.〗^2) in terms of its length (in ft.)

(B) What is the practical domain of this function?

(C) What is the largest area that the fence could enclose?

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Mathematics
1 year
2021-08-30T13:35:34+00:00
2021-08-30T13:35:34+00:00 1 Answers
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## Answers ( )

Answer:(A)(B) Length varies between 1 and 150

(C) Largest area is 22500ft²

Step-by-step explanation:Suppose length is l and width is w.The rectangular garden has perimeter of 600ft, which is mathematically represented as

Area of a rectangle is calculated as

Now, we have a system of equations:

Isolate w, so we have l:

w = 300 – l

Substitute in the area equation:

A = l(300 – l)

A = 300l – l²

(A)

Function of area in terms of length is given byA = 300l – l²(B) The practical domain for this function is values between 1 and 150.

(C) For the largest area, we need to determine the largest garden possible. For that, we take first derivative of the function:

A’ = 300 – 2l

Find the values of l when A’=0:

300 – 2l = 0

2l = 300

l = 150

Replace l in the equation:

w = 300 – 150

w = 150

Now, calculate the largest area:

A = 150*150

A = 22500

The largest area the fence can enclose is22500ft².