How would you answer this math geometric question

* The shorter side is 270 feet

* The longer side is 540 feet

* The greatest possible area is 145800 feet²

Step-by-step 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

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step-by-step explanation: