Question

What is the length of one cycle of the cardioid: `r = 5+5cos (theta)`?

Answer 1

Length of one cycle `=40" "`units

Explanation:

I consider this length of arc using a formula from Calculus with given polar curve equation

`r=5+5 cos theta`

`L=intds`

`ds=sqrt(r^2+((dr)/(d theta))^2) d theta`

For one cycle

`L=2*int_0^pi ds=2*int_0^pi sqrt(r^2+((dr)/(d theta))^2) d theta`

`L=2*int_0^pi sqrt((5+5 cos theta)^2+(-5 sin theta)^2) d theta`

`L=2*int_0^pi sqrt((25+50 cos theta+25 cos^2 theta+25 sin^2 theta)) d theta`

`L=2*int_0^pi sqrt((50+50 cos theta)) d theta`

`L=2sqrt50*int_0^pi sqrt((1+cos theta)) d theta`

`L=40" "`units

graph{(x^2+y^2-5x)^2-25(x^2+y^2)=0[-20,20,-10,10]}

God bless....I hope the explanation is useful.