Suppose that 600 meters of fencing are used to enclose a corral in the
shape of a rectangle on three sides, and then a semicircle on the fourth
side (The diameter of the semicircle is equal to the width of the rectangle).
Find the dimensions of the corral with maximum area.
To find the dimensions of the corral with maximum area, we need to maximize the area function and find the values of the variables that give us the maximum value.
Let's break down the problem into two parts - the rectangle and the semicircle.
1. Rectangle:
Let the length of the rectangle be L and the width be W.
Perimeter of the rectangle = 2L + W (since we are enclosing the corral on three sides)
Perimeter = 600 meters
So, 2L + W = 600
2. Semicircle:
The diameter of the semicircle is equal to the width of the rectangle, so the radius (r) of the semicircle is W/2. The perimeter of the semicircle is the curved part of the fourth side.
Perimeter of the semicircle = πr + r
Perimeter of the semicircle = π(W/2) + (W/2) = (π + 1)(W/2)
Since we are enclosing the corral on three sides, the total perimeter is equal to 600 meters. Therefore, 2L + (π + 1)(W/2) = 600
Now, let's solve the two equations together to find the values of L and W that maximize the area.
1. From the equation 2L + W = 600, we can rewrite it as W = 600 - 2L.
2. Substitute this value of W in the second equation:
2L + (π + 1)((600 - 2L)/2) = 600
2L + (π + 1)(300 - L) = 600
2L + (π + 1)(300) - (π + 1)(L) = 600
2L + (π + 1)(300) - (π + 1)(L) = 600
Expanding it:
2L + 300π + 300 - Lπ - L = 600
Lπ - L + 300 + 300π = 600
(Lπ - L) + (300π + 300) = 600
L(π - 1) + 300(π + 1) = 600
L(π - 1) = 600 - 300(π + 1)
L(π - 1) = 600 - 300π - 300
L(π - 1) = 300 - 300π
L = (300 - 300π)/(π - 1)
L = 100(3 - 3π)/(π - 1)
Now, substitute the value of L back into the equation 2L + W = 600 to find W.
2(100(3 - 3π)/(π - 1)) + W = 600
200(3 - 3π)/(π - 1) + W = 600
W = 600 - 200(3 - 3π)/(π - 1)
Therefore, the dimensions of the corral with maximum area are L = 100(3 - 3π)/(π - 1) and W = 600 - 200(3 - 3π)/(π - 1).