Answer:* The shorter side is 270 feet* The longer side is 540 feet* The greatest possible area is 145800 feet²Step-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