Given that a sports arena will have a 1400 meter perimeter and will have semi-
circles at the ends with a possible rectangular area between the semi-circles, determine
the dimensions of the rectangle and semi-circles that will maximize the total area.
Let the radius of the end sem-circles be r
making the width of the rectangle to be 2r
let the length of the rectangle be x
The perimeter of the field is 2x + 2πr
= 1400
x + πr = 700
x = 700-πr
area = πr^2 + 2rx
= πr^2 + 2r(700-πr)
= πr^2 + 1400r - 2πr^2
d(area)/dr = 2πr + 1400 - 4πr = 0 for a max of area
1400 - 2πr = 0
2πr = 1400
r = 700/π
then x = 700 - π(700/π) = 0
Unexpected strange result!
BUT there is nothing wrong with the calculations, it is the wording of the question that is flawed.
Of course the largest area of any region with a given perimeter is a circle, which my solution has shown.
To determine the dimensions of the rectangle and semi-circles that will maximize the total area, we can set up an equation and find the dimensions that satisfy it.
Let's assume the width of the rectangle is "x" meters. Since there is a semi-circle at each end, the radius of each semi-circle would be "x/2".
The total perimeter of the arena is given as 1400 meters, which includes the perimeter of the rectangle and the perimeters of both semi-circles.
The perimeter of the rectangle is simply the sum of its four sides, which is 2x + 2x = 4x.
The perimeter of a semi-circle is the half-circumference, which is 0.5 * 2πr = πr. Therefore, the combined perimeter of the two semi-circles is 2π(x/2) = πx.
Setting up the equation:
Perimeter of rectangle + Perimeter of semi-circles = 1400
4x + πx = 1400
To find the value of x, we can solve this equation using algebra.
Combining like terms:
(4 + π)x = 1400
Dividing both sides by (4 + π):
x = 1400 / (4 + π)
Now that we have the value of x, we can find the dimensions of the rectangle and semi-circles.
The width of the rectangle is x, and since the length of the rectangle will be determined by the distance between the semi-circles, the length will be (1400 - 4x) / 2.
The radius of each semi-circle is x/2.
Therefore, the dimensions of the rectangle and semi-circles that maximize the total area can be expressed as follows:
Rectangle dimensions: Length = (1400 - 4x) / 2, Width = x
Semi-circle dimensions: Radius = x/2
To find the maximum total area, we can substitute these dimensions into the area formulas for a rectangle and two semi-circles, and then sum them up:
Total Area = Area of Rectangle + Area of Semi-circles
Total Area = (Length * Width) + (2 * π * Radius^2)
Total Area = ((1400 - 4x) / 2 * x) + (2 * π * (x/2)^2)
Now, you can simplify the equation, take the derivative, and find the maximum area by solving for x.