Question

The polar equation r = 5sin(4θ) graphs as a rose.What is the length of the petals of this rose?

Answer 1

we have the polar equation

`r=5sin(4\theta)`

using a graphing tool

The arc length is given by the formula

`L=\int_a^b\sqrt{(r^(\prime)(\theta)^2+(r(\theta))^2}\text{ }d\theta`

where

Find out r' (a derivative of r)

`r^(\prime)(\theta)=20cos(4\theta)`

a=0

b=pi/4

substitute given values in the formula

`\begin{gathered} L=\int_0^{(pi)/(4)}√((20cos(4\theta))^2+(5sin(4\theta))^2)\text{d}\theta \\ L=\int_0^{(pi)/(4)}√(400cos^2(4\theta)+25sin^2(4\theta))\text{d}\theta \end{gathered}`

Solve the integral

The answer is

one minute, please

The arc length is L=10.723 units (three decimal places)