To fill in the blanks with the correct numerical relationships between the volumes of cylinders, cones, and spheres given the same height and radius, we can use the following volume formulas:
- Volume of a cylinder: \( V_{cylinder} = \pi r^2 h \)
- Volume of a cone: \( V_{cone} = \frac{1}{3} \pi r^2 h \)
- Volume of a sphere: \( V_{sphere} = \frac{4}{3} \pi r^3 \)
Let’s express these relationships mathematically for the identical dimensions (i.e., the height \( h \) and radius \( r \) are the same):
A. One sphere + cone(s) = one cylinder
- From the formulas:
\[ V_{sphere} + V_{cone} = V_{cylinder} \] \[ \frac{4}{3} \pi r^3 + \frac{1}{3} \pi r^2 h = \pi r^2 h \] - With \( h = r \): \[ \frac{4}{3} \pi r^3 + \frac{1}{3} \pi r^3 = \pi r^3 \]
- Thus, one sphere + 2 cones = one cylinder.
Fill in A: 2
B. (number of) cone(s) = one cylinder
- As per the relationship:
\[ V_{cone} = \frac{1}{3} V_{cylinder} \] - Therefore, 3 cones = one cylinder.
Fill in B: 3
C. (number of) cone(s) = one sphere
- Using the volume formulas and the fact that:
\[ V_{sphere} = 2 \times V_{cone} \] - Thus, 2 cones = one sphere.
Fill in C: 2
D. One sphere = (number of) cylinder
- As we already established the relationship:
- Since a sphere is equal to 1 cone and equal to 2 cones for being equal to 1 cylinder, one sphere is not equal to one cylinder. We can derive: \[ V_{sphere} = \frac{2}{3} V_{cylinder} \]
- Thus, one sphere = (1.5) cylinder.
Fill in D: \(1.5\)
E. One cone = (number of) cylinder
- From \( 1 cone = \frac{1}{3} cylinder \):
- Therefore, one cone = \( \frac{1}{3} \) cylinder.
Fill in E: \( \frac{1}{3} \)
In summary, the completed blanks are:
A. 2
B. 3
C. 2
D. 1.5
E. 1/3