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.