Question

activity 1 hemispheres 2 what if we pour both a cone and a hemisphere into the cylinder? let’s see what fraction of the cylinder will be filled. describe how the three volumes are related.

Explanation:

Step1: Recall Volume Formulas

Assume the cone, cylinder, and hemisphere have the same radius \( r \) and height \( h \) (for cone and cylinder, \( h = r \) for hemisphere's radius - height relation).

• Volume of cone: \( V_{cone} = \frac{1}{3}\pi r^2 h \)
• Volume of cylinder: \( V_{cylinder} = \pi r^2 h \)
• Volume of hemisphere: \( V_{hemisphere} = \frac{2}{3}\pi r^3 \). If \( h = r \) (cylinder height = radius), \( V_{cylinder} = \pi r^3 \), \( V_{cone} = \frac{1}{3}\pi r^3 \), \( V_{hemisphere} = \frac{2}{3}\pi r^3 \).

Step2: Sum Cone and Hemisphere Volumes

\( V_{total} = V_{cone} + V_{hemisphere} = \frac{1}{3}\pi r^3 + \frac{2}{3}\pi r^3 \)
Simplify: \( V_{total} = \pi r^3 \), which equals \( V_{cylinder} \) (since \( V_{cylinder} = \pi r^2 h \) and \( h = r \), so \( V_{cylinder} = \pi r^3 \)).

Answer:

The sum of the volume of the cone and the hemisphere equals the volume of the cylinder (when they have the same radius and the cylinder’s height equals its radius). So pouring both into the cylinder fills it completely (fraction = 1). The volumes relate as \( V_{\text{cone}} + V_{\text{hemisphere}} = V_{\text{cylinder}} \) (for equal radius \( r \) and cylinder height \( h = r \)).