Geometry Function 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

First draw the figure then it will be more visible..

let the radius of the semi circle be r.

and length of the semicirclr =∏ r

 

So, its diameter     = 2r = breadth of the circle

length of the circle = 3 times radius of the circle

                            = 3r

and also the length of the side of the equilateral triangle = breadth of the triangle

                                                                                  = 2r

 

a)  perimeter of the farm = length of the boundary = 2r +2r + 3r+ 3r + ∏ r

                                                                          = 10r + ∏ r

                                                                          = (10 + ∏) r

 

b) area of the farm = area of triangle + area of rectangle + area of semicircle

                            = (√3/4) side²  + length * breadth +  ( ∏ r²) / 2

                            =  (√3/4) (2r)² + (3r)*(2r) +( ∏ r²) / 2

                            =  (√3/4)*4 r² + 6 r² + ( ∏ r²) / 2

                            =  √3 r² + 6 r² + (∏/2)* r²    

                            = [ √3 + 6+ (∏/2)] * r²  

 

c) area of the farm in terms of the perimeter of the farm = [ √3 + 6+ (∏/2)] * r²  

                                                                                    =  [ √3 + 6+ (∏/2)] * [perimeter/(10+∏)]²