Justin wants to use 188 ft of fencing to fence off the greatest possible rectangular area for a garden. What dimensions should he use? What will be the area of the garden?
A square will encompass the largest area.
188/4 = 47
47 * 47 = 2,209 square feet
Wouldn't 47*47 be incorrect since that would be a square? While the question states the desire for a rectangle. So wouldn't the best choice be 46x48 since it is the closest to 47x47 but it will be a recatangle rather than a square.
To find the dimensions that will result in the greatest possible rectangular area, you can use calculus. However, I will explain a simpler and more intuitive approach using algebra.
Let's represent the length of the garden as 'L' and the width as 'W'. Since the garden is rectangular, we know that the perimeter is given by:
Perimeter = 2(L + W)
In this case, the perimeter is given as 188 ft, so we can write the equation as:
188 = 2(L + W)
We can rearrange the equation to solve for one variable in terms of the other. Let's solve for L:
L = 94 - W
Now, the area of the rectangular garden is given by:
Area = Length * Width = L * W = (94 - W) * W
To find the dimensions that will maximize the area, we can take the derivative of the area function with respect to W and set it equal to zero. Let's take the derivative:
d(Area)/dW = 94 - 2W
Setting the derivative equal to zero and solving for W:
94 - 2W = 0
2W = 94
W = 47
Now that we have the width, we can substitute this value back into the equation for L:
L = 94 - W
L = 94 - 47
L = 47
So the dimensions of the garden that will result in the greatest possible area are 47 ft (length) and 47 ft (width).
To calculate the area, we simply multiply the length and width:
Area = Length * Width = 47 * 47 = 2209 sq ft.
Therefore, the area of the garden will be 2209 square feet.