# What is the distance between the following polar coordinates?:  (21,(10pi)/3), (11,(5pi)/8)

Nov 17, 2017

$29.04 \textcolor{w h i t e}{88}$units

#### Explanation:

First we need to convert the polar coordinates to Cartesian coordinates. We can do this by using the following:

$x = r \cos \left(\theta\right)$

$y = r \sin \left(\theta\right)$

$\therefore$

$x = 21 \cos \left(\frac{10 \pi}{3}\right) = - \frac{21}{2}$

$y = 21 \sin \left(\frac{10 \pi}{3}\right) = - \frac{21 \sqrt{3}}{2}$

Cartesian coordinate:

$\left(- \frac{21}{2} , - \frac{21 \sqrt{3}}{2}\right)$

$x = 11 \cos \left(\frac{5 \pi}{8}\right) = - 4.21$

$y = 11 \sin \left(\frac{5 \pi}{8}\right) = 10.16$

Cartesian coordinate:

$\left(- 4.21 , - 10.16\right)$

Next we use the distance formula:

$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

$d = \sqrt{{\left(- \frac{21}{2} + 4.21\right)}^{2} + {\left(- \frac{21 \sqrt{3}}{2} - 10.16\right)}^{2}}$

$\to = 29.04 \textcolor{w h i t e}{88}$units

All results to 2 .d.p.