leaders of an afterschool program are creating a rectangular garden. They only have 120 feet of fencing available. What would the dimensions of the garden be if the builders attached one side of the fence to the building in order to make the area of the garden as large as possible?
divide the fencing equally among the lengths and widths. The garden of size
60 by 30 will give the maximum area
To find the dimensions of the rectangular garden that would maximize its area, we need to understand the constraints and use calculus to optimize the area.
Let's denote the length of the garden as 'L' and the width as 'W'. Since there are three sides of the garden that need fencing (the length plus the two widths), we have:
L + 2W = 120. (Equation 1)
Furthermore, we want to maximize the area of the rectangular garden, which is given by:
Area = L * W.
To proceed, we'll solve Equation 1 for L and substitute it into the equation for the area:
L = 120 - 2W.
Area = (120 - 2W) * W.
Now, we can find the maximum area by differentiating the equation for area with respect to W and setting it equal to zero:
d(Area)/dW = 120 - 4W. (Differentiation)
120 - 4W = 0. (Setting derivative equal to zero)
Solving the above equation, we find W = 30.
Substituting the value of W back into Equation 1, we find:
L + 2(30) = 120,
L + 60 = 120,
L = 60.
Therefore, the dimensions of the rectangular garden that would maximize its area are 60 feet in length and 30 feet in width.
To make the area of the garden as large as possible, the rectangular garden should be a square.
Given that the leaders have 120 feet of fencing available, they would use three sides of the fence to enclose the garden, while attaching the fourth side to the building.
Since a square has all sides of equal length, we can divide the available 120 feet of fencing by 3 to get the length of each side.
120 feet / 3 = 40 feet
Therefore, the dimensions of the garden would be 40 feet by 40 feet.