You have 100 feet of fencing and decide to make a rectangular garden. What is the largest area enclosed
1 answer:
Answer:
625 ft^2
Step-by-step explanation:
Given
--- perimeter
Required
The largest area
The perimeter is calculated as:
![P = 2 * ( L + W)](https://tex.z-dn.net/?f=P%20%3D%202%20%2A%20%28%20L%20%2B%20W%29)
So, we have:
![2 * ( L + W) = 100](https://tex.z-dn.net/?f=2%20%2A%20%28%20L%20%2B%20W%29%20%3D%20100)
Divide both sides by 2
![L + W = 50](https://tex.z-dn.net/?f=L%20%2B%20W%20%20%3D%2050)
Make L the subject
![L = 50 - W](https://tex.z-dn.net/?f=L%20%3D%2050%20-%20W)
The area is calculated as:
![A= L * W](https://tex.z-dn.net/?f=A%3D%20L%20%2A%20W)
Substitute ![L = 50 - W](https://tex.z-dn.net/?f=L%20%3D%2050%20-%20W)
![A= (50 - W) * W](https://tex.z-dn.net/?f=A%3D%20%2850%20-%20W%29%20%2A%20W)
Open bracket
![A = 50W - W^2](https://tex.z-dn.net/?f=A%20%3D%2050W%20-%20W%5E2)
Differentiate with respect to W
![A' = 50 -2W](https://tex.z-dn.net/?f=A%27%20%3D%2050%20-2W)
Set to 0; to get the maximum value of W
![50 - 2W = 0](https://tex.z-dn.net/?f=50%20-%202W%20%3D%200)
Collect like terms
![-2W = -50](https://tex.z-dn.net/?f=-2W%20%3D%20-50)
Divide by -2
![W = 25](https://tex.z-dn.net/?f=W%20%3D%2025)
So, the maximum area is:
![A = 50W - W^2](https://tex.z-dn.net/?f=A%20%3D%2050W%20-%20W%5E2)
![A = 50 * 25 - 25^2](https://tex.z-dn.net/?f=A%20%3D%2050%20%2A%2025%20-%2025%5E2)
![A = 1250 - 625](https://tex.z-dn.net/?f=A%20%3D%201250%20-%20625)
![A = 625](https://tex.z-dn.net/?f=A%20%3D%20625)
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Step-by-step explanation:
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I = p*r*t
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