A cow is tied to a silo with radius #r# by a rope just long enough to reach the opposite side of the silo. Find the grazing area available for the cow?
Edit: I found this, would that work in solving the problem?
Edit: I found this, would that work in solving the problem?
1 Answer
Explanation:
First, we need to know the length of rope, and we know the rope can reach the opposite side of the silo.
By letting the length of the rope be
This picture depicts the area the cow can graze and ignore the
Now to find the area we shall split it into 2 portions,

The semicircle on the left of the diagram

The circle involute on the right of the diagram
1. To find the semicircle,
Find area of the semicircle
#A_1=1/2*pi(pir)^2#
#color(white)(a)=1/2pi^3r^2#
2. To find the circle involute,
Find the length of wrapped rope
#s# ,
#s=rtheta# Find the length of unwrapped rope
#r# ,
#r=ls#
#color(white)(r)=pirrtheta#
#color(white)(r)=r(pitheta)# Find the area of the circle involute,
#A_2=1/2int_0^pir^2d theta#
#color(white)(A_2)=1/2int_0^pi r^2(pitheta)^2 d theta#
#color(white)(A_2)=r^2/2[(pitheta)^3/3]_0^pi#
#color(white)(A_2)=r^2/6[(pipi)^3(pi0)^3]#
#color(white)(A_2)=(pi^3r^2)/6#
#color(white)(A_2)=(pi^3r^2)/6#
Now, we can find the total area,
Therefore, the grazing area for the cow is