Question

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

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

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

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

\]

Step3: Calculate the volume of the cylinder

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

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

\]

Step4: Calculate the volume of the hemisphere

The formula for the volume of a sphere is \( \frac{4}{3}\pi r^3 \), so the volume of a hemisphere is \( \frac{2}{3}\pi r^3 \). Here, \( r = 6 \) cm.
\[

$$\begin{align*} V_{\text{hemisphere}} &= \frac{2}{3} \times 3.1416 \times 6^3 \\ &= \frac{2}{3} \times 3.1416 \times 216 \\ &= 3.1416 \times 144 \\ &= 452.1696 \end{align*}$$

\]

Step5: Calculate the total volume

Add the volumes of the cone, cylinder, and hemisphere.
\[

$$\begin{align*} V_{\text{total}} &= 339.2928 + 2035.752 + 452.1696 \\ &= 2827.2144 \end{align*}$$

\]
Rounding to two decimal places, \( V_{\text{total}} \approx 2827.21 \)

Answer:

\( V_{\text{cone}} = 339.29 \) \( \text{cm}^3 \)
\( V_{\text{cylinder}} = 2035.75 \) \( \text{cm}^3 \)
\( V_{\text{hemisphere}} = 452.17 \) \( \text{cm}^3 \)
\( V_{\text{total}} = 2827.21 \) \( \text{cm}^3 \)

(Note: The slight differences in the hemisphere and total volume are due to rounding at each step. If we round only at the end, the hemisphere volume is 452.1696 ≈ 452.17, and the total is 339.2928 + 2035.752 + 452.1696 = 2827.2144 ≈ 2827.21)