Which conical container holds more water? Cone 1: height: 8 Diamiter: 4 Cone 2: Height 8 Diamiter 6 Figure not drawn to scale.

1 answer

To compare the volumes of the two cones, we can use the formula for the volume of a cone:

\[ V = \frac{1}{3} \pi r^2 h \]

where \( r \) is the radius and \( h \) is the height.

Cone 1

  • Height (\( h_1 \)): 8
  • Diameter: 4 → Radius (\( r_1 \)): \( \frac{4}{2} = 2 \)

Calculating the volume of Cone 1:

\[ V_1 = \frac{1}{3} \pi (2)^2 (8) \] \[ V_1 = \frac{1}{3} \pi (4)(8) \] \[ V_1 = \frac{32}{3} \pi \]

Cone 2

  • Height (\( h_2 \)): 8
  • Diameter: 6 → Radius (\( r_2 \)): \( \frac{6}{2} = 3 \)

Calculating the volume of Cone 2:

\[ V_2 = \frac{1}{3} \pi (3)^2 (8) \] \[ V_2 = \frac{1}{3} \pi (9)(8) \] \[ V_2 = \frac{72}{3} \pi \] \[ V_2 = 24 \pi \]

Comparing Volumes

Now, we have:

  • Volume of Cone 1: \( \frac{32}{3} \pi \)
  • Volume of Cone 2: \( 24 \pi \)

To make the comparison easier, we can express \( 24 \pi \) with the same denominator as \( \frac{32}{3} \pi \):

\[ 24 \pi = \frac{72}{3} \pi \]

Now it is easier to compare:

  • Cone 1: \( \frac{32}{3} \pi \)
  • Cone 2: \( \frac{72}{3} \pi \)

Clearly, since \( \frac{72}{3} > \frac{32}{3} \), Cone 2 holds more water.

Conclusion

Cone 2 holds more water.