To analyze the volumes of a sphere, cylinder, and cone with the given dimensions, we can use the following formulas:
- Volume of a sphere: \( V = \frac{4}{3} \pi r^3 \)
- Volume of a cylinder: \( V = \pi r^2 h \)
- Volume of a cone: \( V = \frac{1}{3} \pi r^2 h \)
Given:
- The radius \( r \) is the same for all three shapes.
- The height \( h \) of the cylinder and cone is \( 2r \).
Calculating the volumes:
-
Volume of the cylinder: \[ V_{cylinder} = \pi r^2 (2r) = 2\pi r^3 \]
-
Volume of the cone: \[ V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
-
Volume of the sphere: \[ V_{sphere} = \frac{4}{3} \pi r^3 \]
Relationships between the volumes:
Now, comparing the volumes:
-
Volume of the cone in terms of the cylinder: \[ V_{cylinder} = 2 V_{cone} \] because \( V_{cone} = \frac{2}{3} \pi r^3 \) hence \( 2 V_{cone} = 2 \times \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3 \) which equals \( V_{sphere} \).
-
From this, we can also see: \[ V_{cylinder} = V_{sphere} + V_{cone} \]
Conclusion:
Thus, the correct relationship among the given options is: \[ 1 , \text{cylinder} = 2 , \text{cones} + 1 , \text{sphere} \]
So, the correct response is:
1 cylinder = 2 cones + 1 sphere.
3 answers