Geometry Question

Question

A farm has the shape of a rectangle with an equilateral triangle on one end of the rectangle and a semicircle on the opposite end of the rectangle. The length of the rectangle is three times the radius of the semicircle.
a.) Write a function for the perimeter of the farm in terms of the radius of the semicircle.
b.) Write a function for the area of the farm in terms of the radius of the semicircle.
c.) Write a function for the area of the farm in terms of the perimeter of the farm.
Thank you!

Answer 1

Hello Claire! For the perimeter we want to add 2x(length of rectangle) + pi*(radius) + 2x(length of side of triangle). We only use 2x( length of rectangle) from the 4 sides of the rectangle because the two widths are joined to the equilateral and semicircle, so we don't include the widths in the perimeter. The problem gives that the length of the rectangle is 3x(radius). Then there are just two sides from the equilateral because one side of the equilateral triangle is joined to one of the widths of the rectangle. And finally half the circumference (recall that circumference = 2*pi*r) corresponds to the part of the circle that is not joined to the rectangle. To finish up, note that 2x(radius) is the width of the rectangle and since the other side of the rectangle is also a side of the equilateral (recall the sides of an equilateral all have the same lengths) we get that length of side of triangle = width of rectangle = 2x(radius). So If we let R=radius of semicircle the function for perimeter is

f(R) = 2(3R)+(pi)(R)+2(2R) = 10R+Rpi