Question

drag each label to the correct location on the image. not all labels will be used. determine the correct dimensions for the given figures so that the volume of the cone is equal to the volume of the cylinder, and the height of the cylinder is times the height of the cone. figures are not drawn to scale. 3 cm 6 cm 18 cm 15 cm 20 cm 25 cm

Explanation:

Step1: Define variables

Let $r_c$ = cone radius, $h_c$ = cone height, $R$ = cylinder radius, $H$ = cylinder height.
Given: $H = \frac{4}{3}h_c$, $V_{cone}=V_{cylinder}$

Step2: Set volume equality

Volume of cone: $V_{cone}=\frac{1}{3}\pi r_c^2 h_c$
Volume of cylinder: $V_{cylinder}=\pi R^2 H$
Set equal: $\frac{1}{3}\pi r_c^2 h_c = \pi R^2 H$
Cancel $\pi$: $\frac{1}{3}r_c^2 h_c = R^2 H$

Step3: Substitute height relation

Substitute $H=\frac{4}{3}h_c$:
$\frac{1}{3}r_c^2 h_c = R^2 \cdot \frac{4}{3}h_c$
Cancel $\frac{1}{3}h_c$ (non-zero): $r_c^2 = 4R^2$ → $r_c = 2R$

Step4: Match given values

We need $r_c=2R$, and $H=\frac{4}{3}h_c$ from the list {3,6,18,15,20,25}:

• If $R=3$ cm, then $r_c=6$ cm
• Check height: $H=\frac{4}{3}h_c$. If $h_c=15$ cm, $H=\frac{4}{3}\times15=20$ cm. Both 15 and 20 are in the list.

This uses the valid values: 3,6,15,20, leaving 18,25 unused.

Answer:

Cone: Radius = 6 cm, Height = 15 cm; Cylinder: Radius = 3 cm, Height = 20 cm
*(Alternative valid pair: Cone: Radius = 3 cm, Height = 20 cm; Cylinder: Radius = 6 cm, Height = 15 cm is invalid as $\frac{15}{20}
eq\frac{4}{3}$, so only the first pair fits the height ratio)*