Question

suppose a sphere and a cone share the same radius and the height of the cone is twice the radius. which ratio describes the relationship between the formulas for the volume of a cone to the volume of a sphere? (a penny)

Explanation:

Step1: Recall volume formulas

The volume of a sphere is given by \( V_{sphere}=\frac{4}{3}\pi r^{3} \), and the volume of a cone is \( V_{cone}=\frac{1}{3}\pi r^{2}h \).

Step2: Substitute the height of the cone

We know that the height of the cone \( h = 2r \) (since the height of the cone is twice the radius, and they share the same radius \( r \)). Substitute \( h = 2r \) into the cone's volume formula: \( V_{cone}=\frac{1}{3}\pi r^{2}(2r)=\frac{2}{3}\pi r^{3} \).

Step3: Find the ratio of cone to sphere volume

Now, find the ratio \( \frac{V_{cone}}{V_{sphere}} \). Substitute the formulas we have: \( \frac{\frac{2}{3}\pi r^{3}}{\frac{4}{3}\pi r^{3}} \). The \( \pi r^{3} \) and \( \frac{1}{3} \) terms cancel out, leaving \( \frac{2}{4}=\frac{1}{2} \). Wait, no, wait: Wait, \( \frac{\frac{2}{3}\pi r^{3}}{\frac{4}{3}\pi r^{3}}=\frac{2}{3}\times\frac{3}{4}=\frac{2}{4}=\frac{1}{2} \)? Wait, no, let's recalculate. Wait, \( \frac{2/3}{4/3}=\frac{2}{3}\times\frac{3}{4}=\frac{2}{4}=\frac{1}{2} \)? Wait, no, that would mean the cone's volume is half the sphere's? Wait, no, maybe I made a mistake. Wait, sphere volume: \( \frac{4}{3}\pi r^{3} \), cone volume: \( \frac{1}{3}\pi r^{2}h \), \( h = 2r \), so cone volume is \( \frac{1}{3}\pi r^{2}(2r)=\frac{2}{3}\pi r^{3} \). Then the ratio of cone to sphere is \( \frac{2/3\pi r^{3}}{4/3\pi r^{3}}=\frac{2}{4}=\frac{1}{2} \)? Wait, no, \( \frac{2/3}{4/3}=\frac{2}{4}=\frac{1}{2} \), so the ratio of the volume of the cone to the volume of the sphere is \( \frac{1}{2} \)? Wait, no, wait, maybe I mixed up. Wait, let's do it again. Sphere: \( V_s=\frac{4}{3}\pi r^3 \), Cone: \( V_c=\frac{1}{3}\pi r^2 h \), \( h = 2r \), so \( V_c=\frac{1}{3}\pi r^2(2r)=\frac{2}{3}\pi r^3 \). Then \( \frac{V_c}{V_s}=\frac{\frac{2}{3}\pi r^3}{\frac{4}{3}\pi r^3}=\frac{2}{4}=\frac{1}{2} \). Wait, so the ratio of the volume of the cone to the volume of the sphere is \( \frac{1}{2} \)? Wait, no, that seems off. Wait, no, maybe the ratio of sphere to cone? Wait, the question says "the ratio describes the relationship between the formulas for the volume of a cone to the volume of a sphere". So \( \frac{V_{cone}}{V_{sphere}}=\frac{\frac{2}{3}\pi r^{3}}{\frac{4}{3}\pi r^{3}}=\frac{2}{4}=\frac{1}{2} \). Wait, but let's check with numbers. Let \( r = 3 \). Then sphere volume: \( \frac{4}{3}\pi(27)=36\pi \). Cone height: \( h = 6 \). Cone volume: \( \frac{1}{3}\pi(9)(6)=18\pi \). Then \( \frac{18\pi}{36\pi}=\frac{1}{2} \). Yes, so the ratio of the volume of the cone to the volume of the sphere is \( \frac{1}{2} \), or 1:2.

Answer:

The ratio of the volume of the cone to the volume of the sphere is \(\boldsymbol{\frac{1}{2}}\) (or 1:2).