To find the dimensions (r and L) that make the picnic area maximum, we can use calculus and optimization techniques.
Let's assume the length of the rectangle is L meters and the radius of the semicircles is r meters.
The total length of the circuit can be expressed as:
Length_of_Rectangle + 2 * Circumference_of_Semicircle = 2000 meters
The length of the rectangle is L meters, and the width of the rectangle is r meters (since a rectangle's length is the longer side).
Length_of_Rectangle = L meters, Width_of_Rectangle = r meters
The circumference of a semicircle with radius r can be expressed as:
Circumference_of_Semicircle = πr meters
So, the equation becomes:
L + 2Ï€r = 2000 (Equation 1)
Now, let's express the area of the picnic area (rectangle area minus the semicircular areas):
Area_of_Picnic_Area = Length_of_Rectangle * Width_of_Rectangle - 2 * (Area_of_Semicircle)
= L * r - 2 * (Ï€r^2/2)
= L * r - πr^2
To maximize the area, we can find the critical points of the function and check for the maximum. Taking the derivative of the area function with respect to r, we get:
d(Area_of_Picnic_Area)/dr = L - 2Ï€r
The critical point occurs where the derivative is zero. So, setting the derivative equal to zero:
L - 2Ï€r = 0
2Ï€r = L
r = L/(2Ï€) (Equation 2)
Substituting equation 2 back into equation 1:
L + 2Ï€(L/(2Ï€)) = 2000
L + L = 2000
2L = 2000
L = 1000 meters
Now that we have the length of the rectangle, we can find the radius of the semicircles by substituting back into equation 2:
r = 1000 / (2Ï€)
r ≈ 159.154943 meters
Therefore, to make the picnic area maximum, the dimensions are approximately L = 1000 meters and r ≈ 159.154943 meters.