Questions: A gardener has 1160 feet of fencing to fence in a rectangular garden. One side of the garden is bordered by a river and so it does not need any fencing. What dimensions would guarantee that the garden has the greatest possible area? shorter side: square ft (feet) longer side: square ft (feet) greatest possible area: square ft^2 (square-feet)

Solution Steps

Let \( x \) be the length of the side parallel to the river (longer side), and \( y \) be the length of the two sides perpendicular to the river (shorter sides).

Since one side is bordered by the river, the total fencing used is: \[ x + 2y = 1160 \]

Solve for \( x \): \[ x = 1160 - 2y \]

The area \( A \) of the rectangle is: \[ A = x \cdot y \] Substitute \( x \) from the previous step: \[ A = (1160 - 2y) \cdot y \] \[ A = 1160y - 2y^2 \]

To find the maximum area, take the derivative of \( A \) with respect to \( y \) and set it to zero: \[ \frac{dA}{dy} = 1160 - 4y = 0 \] Solve for \( y \): \[ 4y = 1160 \] \[ y = 290 \]

Substitute \( y = 290 \) back into the equation for \( x \): \[ x = 1160 - 2(290) \] \[ x = 1160 - 580 \] \[ x = 580 \]

Substitute \( x = 580 \) and \( y = 290 \) into the area formula: \[ A = 580 \cdot 290 \] \[ A = 168200 \]

Final Answer