Question:
Jackson buys a grape snow cone on a hot day. By the time he eats all the "snow" off the top, the paper cone is filled with 27\pi27π27, pi cm^3 3 start superscript, 3, end superscript of melted purple liquid. The radius of the cone is 333 cm. What is the height of the cone?
Answer:
The height of the cone is ![9 \ cm](https://tex.z-dn.net/?f=9%20%5C%20cm)
Explanation:
It is given that the radius of the cone is ![3 \ cm](https://tex.z-dn.net/?f=3%20%5C%20cm)
The volume of the cone is ![27\pi](https://tex.z-dn.net/?f=27%5Cpi)
The height of the cone can be determine using the formula,
![$V=\frac{1}{3} \pi r^{2} h$](https://tex.z-dn.net/?f=%24V%3D%5Cfrac%7B1%7D%7B3%7D%20%5Cpi%20r%5E%7B2%7D%20h%24)
Substituting the values
and
, we get,
![$27 \pi=\frac{1}{3} \pi(3)^{2} h$](https://tex.z-dn.net/?f=%2427%20%5Cpi%3D%5Cfrac%7B1%7D%7B3%7D%20%5Cpi%283%29%5E%7B2%7D%20h%24)
Multiplying both sides by 3, we have,
![$81 \pi= 9\pi h$](https://tex.z-dn.net/?f=%2481%20%5Cpi%3D%209%5Cpi%20h%24)
Dividing both sides by
, we have,
![9=h](https://tex.z-dn.net/?f=9%3Dh)
Thus, the height of the cone is ![9 \ cm](https://tex.z-dn.net/?f=9%20%5C%20cm)