[highlight]
[b]Frustums[/b]
-Frustums consist of 1 large cone, minus a smaller one
-The smaller cone's radius is proportional to the large cone's radius
-Therefore the smaller cone's height also follows the same ratio and vice versa
-To find the volume of the [b]frustum[/b], calculate the volume of the large cone and find its difference with the small cone[br]
[/highlight]
As you will be needing to find the volume of a cone, you need the following equation:
[code]
([pi]r[squared]h) [divide] 3 [r = radius; h = height]
[/code]
[h3]Example 1[/h3]
Find the volume of [b]A[/b] [b](to 1 decimal place)[/b], when its height is [b]32cm[/b] and its diameter is [b]20cm[/b].
[img src='https://res.cloudinary.com/deylrqt2d/image/upload/v1485040923/Shape-1_yliu76.jpg']
[i]First we need to find the proportion of the smaller [b]blue cone[/b] to the [b]entire cone[/b].[/i] [br]
To do this, first find the total height:
[code]In this case the total height is: [br] 32cm + 8cm = [b]40cm[/b][/code]
([u]To reduce complexity lets label the [i]smaller blue cone[/i] [b]B[/b][/u]).[br]
[highlight]
Try to simplify both sides to either one or less.
[table]
[tr]
[td]Full cone height[/td][td]:[/td][td]Smaller cone height[/td]
[/tr]
[tr]
[td]40[/td][td]:[/td][td]8[/td]
[/tr]
[tr]
[td]4[/td][td]:[/td][td]0.8[/td]
[/tr]
[tr]
[td]1[/td][td]:[/td][td]0.2[/td]
[/tr]
[/table]
[/highlight]
Now we know that the proportion between the heights of the cones is [b]1 : 0.2[/b], we can tell work out the radius of cone [b]B[/b] as well.[br]
[u](Though first we'll need to find the radius of the cone [b]A[/b], which would be [b]10cm[/b] as half the [b]diameter[/b], 20cm, is 10cm)[/u][br]
Now all we need to do, is multiply to use the proportion we found, to find the radius of [b]B[/b].
[highlight]
The proportion that we earlier got means that both the height and radius of [b]B[/b] is [b]0.2[/b] times as big as that of [b]A[/b], or the full cone.[br]
Therefore, just how we get the height of [b]B[/b] by multiplying [b]A[/b]'s height by 0.2, we can do the same for the radius.
[code]
10 [multiply] 0.2 = 2 [10 is the radius of [b]A[/b]]
[/code]
[/highlight]
So the radius of [b]B[/b] is 2.[br][br]
Now we can move on to finding the volume.[br]
First, get the volume of the full cone (so both [b]A[/b] and [b]B[/b]'s combined volume):
[code]
[table]
[tr]
[td] Volume [b]Full Cone[/b][/td][td][space]=[space][/td][td style='text-align:left']([pi]r[squared]h) [divide] 3[/td]
[/tr]
[tr]
[td][/td][td][space]=[space][/td][td style='text-align:left']([pi] [multiply] 10[squared] [multiply] 40) [divide] 3[/td]
[/tr]
[tr]
[td][/td][td][space]=[space][/td][td style='text-align:left'](4000[pi]) [divide] 3[/td]
[/tr]
[tr]
[td][/td][td][space]=[space][/td][td style='text-align:left']1333.33.... [multiply] [pi] [/td]
[/tr]
[tr]
[td][/td][td][space]=[space][/td][td style='text-align:left']4188.790205cm[cubed][/td]
[/tr]
[/table]
(I've left it to [b]7 significant figures[/b], just so I don't lose too much accuracy)
[/code]
[br]
Now we get the volume of [b]B[/b] (the smaller cone):
[code]
[table]
[tr]
[td] Volume of [b]B[/b][/td][td][space]=[space][/td][td style='text-align:left']([pi]r[squared]h) [divide] 3[/td]
[/tr]
[tr]
[td][/td][td][space]=[space][/td][td style='text-align:left'] ([pi] [multiply] 2[squared] [multiply] 8) [divide] 3[/td]
[/tr]
[tr]
[td][/td][td][space]=[space][/td][td style='text-align:left']32[pi] [divide] 3[/td]
[/tr]
[tr]
[td][/td][td][space]=[space][/td][td style='text-align:left']10.666...... [multiply] [pi] [/td]
[/tr]
[tr]
[td][/td][td][space]=[space][/td][td style='text-align:left']33.5103216cm[cubed][/td]
[/tr]
[/table]
(I've also left it to [b]7 significant figures[/b] here, just so I, again, don't lose too much accuracy)
[/code]
[br]
Now all that's left is to get the volume of [b]A[/b]:
[code]
[table]
[tr]
[td] Volume of [b]A[/b][/td][td][space]=[space][/td][td style='text-align:left']([b]Volume of full cone[/b]) - ([b]Volume of B[/b])[/td]
[/tr]
[tr]
[td][/td][td][space]=[space][/td][td style='text-align:left']4188.790205 - 33.5103216[/td]
[/tr]
[tr]
[td][/td][td][space]=[space][/td][td style='text-align:left']4155.2798834[/td]
[/tr]
[tr]
[td][/td][td][space]=[space][/td][td style='text-align:left']4155.3cm[cubed] (to 1 decimal place)[/td]
[/tr]
[/table]
[/code]
[highlight]
[i]Therefore the volume of [b]A[/b] is [b]4155.3cm[cubed] (to 1 decimal place)[/b][/i]
[/highlight]