Answer:
* The shorter side is 270 feet
* The longer side is 540 feet
* The greatest possible area is 145800 feet²
Explanation:
* Lets explain how to solve the problem
- There are 1080 feet of fencing to fence a rectangular garden
- One side of the garden is bounded by a river so it doesn't need
any fencing
- Consider that the width of the rectangular garden is x and its length
is y and one of the two lengths is bounded by the river
- The length of the fence = 2 width + length
∵ The width = x and the length = y
∴ The length of the fence = 2x + y
- The length of the fence = 1080 feet
∴ 2x + y = 1080
- Lets find y in terms of x
∵ 2x + y = 1080 ⇒ subtract 2x from both sides
∴ y = 1080 - 2x ⇒ (1)
- The area of the garden = Length × width
∴ The area of the garden is A = xy
- To find the greatest area we will differentiate the area of the garden
with respect to x and equate the differentiation by zero to find the
value of x which makes the area greatest
∵ A = xy
- Use equation (1) to substitute y by x
∵ y = 1080 -2x
∴ A = x(1080 - 2x)
∴ A = 1080x - 2x²
# Remember
- If y = ax^n, then dy/dx = a(n) x^(n-1)
- If y = ax, then dy/dx = a (because x^0 = 1)
∵ A = 1080x - 2x²
∴ dA/dx = 1080 - 2(2)x
∴ dA/dx = 1080 - 4x
- To find x equate dA/dx by 0
∴ 1080 - 4x = 0 ⇒ add 4x to both sides
∴ 1080 = 4x ⇒ divide both sides by 4
∴ x = 270
- Substitute the value of x in equation (1) to find the value of y
∵ y = 1080 - 2x
∴ y = 1080 - 2(270) = 1080 - 540 = 540
∴ y = 540
* The shorter side is 270 feet
* The longer side is 540 feet
∵ The area of the garden is A = xy
∴ The greatest area is A = 270 × 540 = 145800 feet²
* The greatest possible area is 145800 feet²