Given a cylinder with a diameter that is half as long as its height and a volume of 16,717 cubic units, what is the height to the nearest whole number?

To find the height of the cylinder, we can use the formula for the volume of a cylinder, which is given by:

[ V = \pi r^2 h ]

where ( V ) is the volume, ( r ) is the radius, and ( h ) is the height.

From the problem, we know the volume ( V = 16,717 ) cubic units. The diameter is half as long as the height, which can be expressed mathematically as:

[ d = \frac{h}{2} ]

Since the radius ( r ) is half of the diameter, we can substitute the expression for the diameter to express the radius in terms of height:

[ r = \frac{d}{2} = \frac{h}{4} ]

Now, substituting ( r ) into the volume formula:

[ V = \pi \left(\frac{h}{4}\right)^2 h ]

This simplifies to:

[ V = \pi \left(\frac{h^2}{16}\right) h = \frac{\pi h^3}{16} ]

Setting this equal to the volume given:

[ \frac{\pi h^3}{16