Question

A gardener has 400 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. garden bordered by a river What dimensions would guarantee that the garden has the greatest possible area

Answer 1

To guarantee the greatest possible area for the garden, the gardener should find the dimensions that satisfy the equation L + 2W = 400.

Step-by-step explanation:

To find the dimensions that will guarantee the greatest possible area for the garden, we can use the formula A = L x W, where A is the area, L is the length, and W is the width of the garden.

Let's assume that the river is one of the sides of the garden, so we only need to fence the other three sides.

Since the gardener has 400 feet of fencing, we can set up the following equation: L + 2W = 400.

To maximize the area, we need to solve for L and W. Let's solve the equation for L: L = 400 - 2W.

Substituting this value into the area formula, we get: A = (400 - 2W) x W.

To find the maximum area, we can take the derivative of the area formula and set it equal to zero. Solving this equation will give us the value of W that maximizes the area. We can then find the corresponding value of L using the equation we found earlier.

Therefore, the dimensions that will guarantee the greatest possible area for the garden can be found by solving the equation L + 2W = 400 and finding the values of L and W that satisfy that equation.

Answer 2

The dimensions that would guarantee that the garden has the greatest possible area are 100 ft by 200 ft

What dimensions would guarantee that the garden has the greatest possible area

From the question, we have the following parameters that can be used in our computation:

Perimeter 400 feet

Since one of the garden is bordered by a river, we have

Perimeter = 2x + y

Where x and y are the dimensions

So, we have

2x + y = 400

y = 400 - 2x

The area is calculated as

A = xy

This gives

A = x(400 - 2x)

Expand

A = 400x - 2x²

Differentiate and set to 0

400 - 4x = 0

So, we have

4x = 400

x = 100

Recall that

y = 400 - 2x

y = 400 - 2 * 100

y = 200

Hence, the dimensions are 100 ft by 200 ft