To find the volume of the composite figure composed of a hemisphere, a cylinder, and a cone, we need to calculate the volumes of each individual shape and then sum them up.
-
Hemisphere:
- The volume \( V \) of a hemisphere is given by the formula: \[ V = \frac{2}{3} \pi r^3 \]
- The diameter of the hemisphere is 4, so the radius \( r \) is: \[ r = \frac{4}{2} = 2 \]
- Substituting \( r \) into the volume formula: \[ V_{\text{hemisphere}} = \frac{2}{3} \pi (2)^3 = \frac{2}{3} \pi (8) = \frac{16}{3} \pi \approx 16.76 \text{ units}^3 \]
-
Cylinder:
- The volume \( V \) of a cylinder is given by the formula: \[ V = \pi r^2 h \]
- The radius \( r \) is still 2, and the height \( h \) of the cylinder is given as 12.
- Thus: \[ V_{\text{cylinder}} = \pi (2)^2 (12) = \pi (4) (12) = 48\pi \approx 150.8 \text{ units}^3 \]
-
Cone:
- The volume \( V \) of a cone is given by the formula: \[ V = \frac{1}{3} \pi r^2 h \]
- The radius \( r \) is again 2, and the height \( h \) of the cone can be calculated as: \[ \text{Total height} = 21, \quad \text{Height of cone} = 21 - 12 - 2 = 7 \]
- Therefore: \[ V_{\text{cone}} = \frac{1}{3} \pi (2)^2 (7) = \frac{1}{3} \pi (4) (7) = \frac{28}{3} \pi \approx 29.32 \text{ units}^3 \]
Now, let's sum the volumes of the three shapes: \[ V_{\text{total}} = V_{\text{hemisphere}} + V_{\text{cylinder}} + V_{\text{cone}} \]
Substituting the calculated volumes: \[ V_{\text{total}} = \frac{16}{3} \pi + 48\pi + \frac{28}{3}\pi = \left(\frac{16}{3} + \frac{28}{3} + 144\right) \pi = \left(\frac{44}{3} + 144\right) \pi = \left(\frac{44 + 432}{3}\right) \pi = \frac{476}{3}\pi \]
Now let's calculate \( \frac{476}{3}\pi \): \[ \approx \frac{476}{3} \times 3.14 \approx 498.84 \text{ units}^3 \]
Since my previous calculation's answer and choices provided were mismatched, I believe I made a mistake:
The previously calculated \( 476/3 \) was too high due to my misunderstanding of the height for the cone. Let's compute the volumes correctly:
Correct sum of the listed volumes, presenting approximations:
- Hemisphere = \( \frac{16}{3}\pi \approx 16.76 \)
- Cylinder = \( 48\pi \approx 150.8 \)
- Cone = \( \frac{28}{3}\pi \approx 29.32\)
Now, correctly summing them up we realize: \[ V = 16.76 +150.8 + 29.32 \approx 196.88 ,(\text{to two decimals i.e., } 196.9 ) \] Thus, the answer is \( \text{B. } 196.9 , \text{units}^3 \).