Austin is building a rabbit pen with 25 feet of fence. What are the dimensions of the rectangle hr should build to have the greatest possible area
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To find the dimensions of the rectangle with the greatest possible area, we need to determine its length and width. Let's break down the problem step by step:
Let's say the length of the rectangle is L and the width is W. The perimeter of the rectangle (which is the sum of all four sides) is given as 25 feet. Therefore, we can write the equation:
2L + 2W = 25
Simplifying this equation, we can divide both sides by 2:
L + W = 12.5
To find the rectangle with the greatest possible area, we need to maximize the area function A(L, W) = L * W.
Now, we have two variables and one equation. To find the dimensions of the rectangle that maximize the area, let's assume a value for one variable and solve for the other.
Let's assume W = 12.5 - L (from the previous equation).
Substituting this into the area function, we can write:
A(L) = L * (12.5 - L)
Expanding this equation, we get:
A(L) = 12.5L - L^2
To maximize the area, we need to find the maximum of this quadratic function.
To do this, we differentiate A(L) with respect to L, set it equal to zero, and solve for L:
dA(L)/dL = 12.5 - 2L = 0
Solving this equation, we find:
2L = 12.5
L = 6.25
Now, we can substitute this value of L back into the equation W = 12.5 - L:
W = 12.5 - 6.25
W = 6.25
Therefore, the dimensions of the rectangle with the greatest possible area are:
Length (L) = 6.25 feet
Width (W) = 6.25 feet