Question

calculate the volume of the composite shape shown above.
use 3.1416 for the approximate value of ( pi ). round your answer to two decimal places, as needed.
( v_{\text{cone}} = square , \text{cm}^3 )
( v_{\text{cylinder}} = square , \text{cm}^3 )
( v_{\text{hemisphere}} = square , \text{cm}^3 )
( v_{\text{total}} square , \text{cm}^3 )

Explanation:

Step1: Find the radius of the sphere, cylinder, and cone

The diameter is 12 cm, so the radius \( r = \frac{12}{2} = 6 \) cm.

Step2: Calculate the volume of the cone (\( V_{cone} \))

The formula for the volume of a cone is \( V_{cone} = \frac{1}{3}\pi r^2 h \), where \( h = 9 \) cm, \( r = 6 \) cm, and \( \pi = 3.1416 \).
\[

$$\begin{align*} V_{cone} &= \frac{1}{3} \times 3.1416 \times 6^2 \times 9 \\ &= \frac{1}{3} \times 3.1416 \times 36 \times 9 \\ &= 3.1416 \times 12 \times 9 \\ &= 3.1416 \times 108 \\ &= 339.2928 \end{align*}$$

\]

Step3: Calculate the volume of the cylinder (\( V_{cylinder} \))

The formula for the volume of a cylinder is \( V_{cylinder} = \pi r^2 h \), where \( h = 18 \) cm, \( r = 6 \) cm, and \( \pi = 3.1416 \).
\[

$$\begin{align*} V_{cylinder} &= 3.1416 \times 6^2 \times 18 \\ &= 3.1416 \times 36 \times 18 \\ &= 3.1416 \times 648 \\ &= 2035.752 \end{align*}$$

\]

Step4: Calculate the volume of the hemisphere (\( V_{hemisphere} \))

The formula for the volume of a hemisphere is \( V_{hemisphere} = \frac{2}{3}\pi r^3 \), where \( r = 6 \) cm, and \( \pi = 3.1416 \).
\[

$$\begin{align*} V_{hemisphere} &= \frac{2}{3} \times 3.1416 \times 6^3 \\ &= \frac{2}{3} \times 3.1416 \times 216 \\ &= 3.1416 \times 144 \\ &= 452.3896 \end{align*}$$

\]

Step5: Calculate the total volume (\( V_{total} \))

\[

$$\begin{align*} V_{total} &= V_{cone} + V_{cylinder} + V_{hemisphere} \\ &= 339.2928 + 2035.752 + 452.3896 \\ &= 2827.4344 \end{align*}$$

\]

Answer:

\( V_{cone} = 339.29 \) \( \text{cm}^3 \) (rounded to two decimal places)
\( V_{cylinder} = 2035.75 \) \( \text{cm}^3 \) (rounded to two decimal places)
\( V_{hemisphere} = 452.39 \) \( \text{cm}^3 \) (rounded to two decimal places)
\( V_{total} = 2827.43 \) \( \text{cm}^3 \) (rounded to two decimal places)