SOLUTION: Alan is building a garden shaped like a rectangle with a semicircle attached to one short side. If he has 40 feet of fencing to go around it, what dimensions will give him the maxi

Question 288478: Alan is building a garden shaped like a rectangle with a semicircle attached to one short side. If he has 40 feet of fencing to go around it, what dimensions will give him the maximum area in the garden?
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
Alan is building a garden shaped like a rectangle with a semicircle attached to one short side.
If he has 40 feet of fencing to go around it, what dimensions will give him the maximum area in the garden?
:
Short side = W
The half circumference of the semi circle = .5*pi*W
:
2Lengths + width + semicircle = 40 ft
2L + W + (.5*pi*W) = 40
2L + W + 1.57W = 40
Combine like terms
2L + 2.57W = 40
Simplify divide by 2
L + 1.285W = 20
L = (20-1.285W); for substitution
:
Area equation
Rectangle area + semicircle area
A = L * W + (.5*pi*(.5W)^2)
A = LW + (1.57*.25W^2)
A = LW + .3925W^2
Replace L with (20-1.285W)
A = W(20-1.285W) + .3925W^2
A = -1.285W^2 + .3925W^2 + 20W
A = -.8925W^2 + 20W
:
Find W by finding the axis of symmetry of this equation a=-.8925, b=20
W =
W =
W = +11.2 ft is the width for max area
then
L = 20 - 1.285(11.2)
L = 20 - 14 = 5.6 ft is the length
:
This presents a problem, the semi circle is supposed to be on the short side.
These calculations say the max area will be when the semi circle is on the long side.
:
Check the perimeter
2(5.6) + 11.2 + 1.57(11.2) =
11.2 = 11.2 + 17.6 = 40
:
I can't see how we could do it any other way
:
Rectangle: 5.6 by 11.2, semicircle circumference 17.6
:
Max area: (5.6*11.2) + (.5*pi*5.6^2) = 112 sq/ft
RELATED QUESTIONS
Alan is building a garden shaped like a rectangle with a semicircle attached to one short (answered by stanbon)

A patio needs to be painted. The shape of the patio is made up of a rectangle with a... (answered by Cromlix)

Another customer wants a garden with the same perimeter. If the second garden is shaped... (answered by solver91311)

Connor has a garden shaped like a rectangle with dimensions of 12' by 30'. How many feet... (answered by checkley77)

Ol' MacDonald has a farm lot that is shaped like a rectangle 40 feet wide and 50 feet... (answered by sudhanshu_kmr)

A driveway 16 feet wide is shaped like a semicircle. Determine the area of the driveway. (answered by Cromlix)

A semicircle is attached on one side of a rectangle. The length of the rectangle is 10cm... (answered by josgarithmetic)

3. A man wishes to have a rectangular shaped garden in his backyard. He has 84 feet of... (answered by ikleyn,josmiceli)

Bob has 40 feet of fencing to enclose a rectangular garden. If one side of the garden is... (answered by josgarithmetic)