An indoor physical fitness room consists of a rectangular region with a semicircle on each end. The perimeter of the room is to be a 200m running track. Find the dimension that will make the area of the rectangular as large as possible.
Perimeter=2L + PI*W think about that.
Area=LW+PI*w^2/4
L from the first equation is (Perimeter-PI/W)1/2
Put that value of L in the second area equation, take the derivative of area with respect to W, set to zero, and solve for W. Then go back to the perimeter equation and solve for L.
To find the dimensions that will make the area of the rectangular region as large as possible, we need to set up an equation representing the perimeter of the room.
Let's assume that the length of the rectangular region is "L" and the width is "W". Since there are two semicircles on each end, their combined circumference is equal to the length of the running track, which is 200m.
The formula to calculate the circumference of a semicircle with radius "r" is C = πr + 2r, where π is a mathematical constant approximately equal to 3.14.
So, the equation representing the perimeter is:
2L + πr + 2r = 200
Now, we need to isolate either "L" or "W" in order to express the area of the rectangle in terms of one variable. Let's isolate "L" by subtracting the terms involving "r" from both sides of the equation:
2L = 200 - πr - 2r
2L = 200 - (π + 2)r
L = (200 - (π + 2)r)/2
The area of a rectangle is given by the formula A = L × W. Substituting for L, we get:
A = (200 - (π + 2)r)/2 × W
A = (200 - (π + 2)r)W/2
To find the maximum area, we need to differentiate this equation with respect to "r" and set it equal to zero. However, since solving a derivative equation may be complex, we can solve it using optimization techniques.
Since the perimeter of the room must be a 200m running track, we can substitute for L in terms of r:
2L + πr + 2r = 200
2(200 - (π + 2)r)/2 + πr + 2r = 200
200 - (π + 2)r + πr + 2r = 200
200 - πr - 2r + πr + 2r = 200
r = 200/(π + 4)
Now substitute this value of r back into the equation for L:
L = (200 - (π + 2)r)/2
L = (200 - (π + 2) * 200/(π + 4))/2
Calculating this equation will give us the value of L. Furthermore, to determine the dimensions that make the area of the rectangle as large as possible, substitute the values of L and r into the area equation:
A = (200 - (π + 2) * 200/(π + 4))/2 * W
Calculating this will give us the maximum area of the rectangular region.