Determine the maximum area of a piece of land comprising a semi-circle adjacent to a rectangle that can be enclosed by 850 metres of fencing.
A = (1/2) pi r^2 + 2 r w
perimeter = 2 r + 2 w + 2 pi r = 850
= 2 r(1+pi) + 2 w
425 = r(4.14) + 2 w
213 = 2.07 r + w
w = (213 - 2.07 r)
A = .5 pi r^2 + 2r(213 - 2.07r)
A = 1.57 r^2 + 425 r - 4.14 r^2
2.57 r^2 -425 r = -A
r^2 - 165 r = -A/2.57
r^2 - 165 r + 6806 = -A/2.57 + 6806
(r-82.5) = -(1/2.57)( A - 17,492)
so I get
r = 2.5
and
area = 17,492
To determine the maximum area of the piece of land, we need to find the dimensions of the rectangle and the radius of the semi-circle that would maximize the area.
Let's assume the rectangle's length is L and width is W, and the radius of the semi-circle is R.
The total length of the fencing is given as 850 metres, which will be used to form the perimeter of the land.
Perimeter of the land = Perimeter of rectangle + Circumference of semi-circle
Perimeter of rectangle = 2 * (Length + Width) = 2L + 2W
Circumference of semi-circle = π * D = π * 2R, where D is the diameter and D = 2R.
Using the given information, we can write the equation:
2L + 2W + π * 2R = 850
Now, we need to express either L or W in terms of R and substitute it back into the equation, so we have only one variable.
Let's express L in terms of R:
L = (850 - 2W - π * 2R) / 2
To find the maximum area, we need to differentiate the area formula with respect to R and set it equal to zero.
Area of the land = Area of rectangle + Area of semi-circle
Area of rectangle = Length * Width = L * W
Area of semi-circle = (π * R^2) / 2
Total Area = L * W + (π * R^2) / 2
Differentiating Total Area with respect to R:
d(Total Area) / dR = (dL / dR) * W + π * R
Setting the derivative equal to zero, we get:
(dL / dR) * W + π * R = 0
(dL / dR) * W = -π * R
(dL / dR) = (-π * R) / W
Substituting the expression for L:
(d(850 - 2W - π * 2R) / dR) * W = -π * R
Simplifying the equation:
-2W - π * 2R = -π * R
-2W = -π * R + π * 2R
-2W = π * R
W = -(π * R) / 2
Now we can substitute W into the equation for L and simplify:
L = (850 - 2W - π * 2R) / 2
L = (850 - 2(-(π * R) / 2) - π * 2R) / 2
L = (850 + π * R - π * 2R) / 2
L = (850 - π * R) / 2
We can now substitute L and W into the equation for Total Area:
Total Area = L * W + (π * R^2) / 2
Total Area = (850 - π * R) / 2 * (-(π * R) / 2) + (π * R^2) / 2
Now we have the area formula in terms of only one variable, R. To find the maximum area, we can take the derivative of the area formula with respect to R, set it equal to zero, and solve for R. Once we have the value of R, we can substitute it back into the area formula to find the maximum area of the piece of land.