Question

warm-up
1 a cone and a cylinder have the same height and radius.
what fraction of the cylinder will be filled by the cone?

Explanation:

Step1: Recall Volume Formulas

The volume of a cylinder is \( V_{cylinder} = \pi r^2 h \), and the volume of a cone is \( V_{cone} = \frac{1}{3}\pi r^2 h \), where \( r \) is the radius and \( h \) is the height. Since the cone and cylinder have the same radius and height, we can find the fraction by dividing the volume of the cone by the volume of the cylinder.

Step2: Calculate the Fraction

Let \( V_{cone} = \frac{1}{3}\pi r^2 h \) and \( V_{cylinder} = \pi r^2 h \). The fraction \( f \) is \( f=\frac{V_{cone}}{V_{cylinder}}=\frac{\frac{1}{3}\pi r^2 h}{\pi r^2 h} \). The \( \pi r^2 h \) terms cancel out, leaving \( f = \frac{1}{3} \).

Answer:

The cone fills \(\frac{1}{3}\) of the cylinder.