# Math Help - Volume

1. ## Volume

A cone is attached to a hemisphere of radius 4 cm. If the total height of the object is 10 cm, find its volume.

2. ## Re: Volume

Let's calculate the volume of the cone and the volume of the hemisphere seperatly and then add them. I will be accurate to 2 points after the decimal point.

Hemisphere: V = (2/3)*pi*r^3 = 128*pi/3 = 134.04

If the height of the entire object is 10, and the radius of the hemisphere is 4, then the height of the cone is 10 - 4 = 6.

Cone: V = (1/3)*pi*r^2*h = 32*pi = 100.53

Vcone + Vhemisphere = 134.04 + 100.53 = 234.57

3. ## Re: Volume

Hello, Farisco!

A cone is attached to a hemisphere of radius 4 cm.
If the total height of the object is 10 cm, find its volume.

Code:
    -         * * *
:     *     :     *
:   *       :       *
:  *        :4       *
:           :
: * - - - - * - - - - *
:  \   4    :    4   /
10   \       :       /
:    \      :      /
:     \     :6    /
:      \    :    /
:       \   :   /
:        \  :  /
:         \ : /
:          \:/
-           *
We have a hemisphere with radius 4.
We have cone with radius 4 and height 6.

You should be able to find the total volume
. . without a calculator and without rounded-off decimals.

A sphere has volume: . $V \:=\:\tfrac{4}{3}\pi r^3$, where $r$ is the radius.

A half-sphere with radius 4 has volume: . $V \:=\:\tfrac{1}{2} \times \tfrac{4}{3}\pi(4^3) \:=\:\frac{128\pi}{3}$

A circular cone has volume: . $V \:=\:\tfrac{\pi}{3}r^2h\;\;(r = \text{radius, }\:h = \text{height})$

A cone with $r = 4,\,h=6$ has volume: . $V \;=\;\tfrac{\pi}{3}(4^2)(6) \:=\:32\pi$

The total volume is: . $\frac{128\pi}{3} + 32\pi \;=\;\boxed{\frac{224\pi}{3}\text{ cm}^3}$

If you want a decimal, now is the time to crank it out:

. . . . $\boxed{2}\,\boxed{2}\,\boxed{4}\;\;\boxed{\times} \;\; \boxed{\pi}\;\;\boxed{\div}\;\;\boxed{3}\;\;\boxed {=}$

and we get: . $\boxed{234.5722515}$