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.
$v_{cone} = \boxed{\quad} \text{ cm}^3$
$v_{cylinder} = \boxed{\quad} \text{ cm}^3$
$v_{hemisphere} = \boxed{\quad} \text{ cm}^3$
$v_{total} = \boxed{\quad} \text{ cm}^3$

Explanation:

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

The diameter of the circular base (for cone, cylinder, and hemisphere) 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 \( V_{\text{cone}} = \frac{1}{3}\pi r^2 h \), where \( h = 3 \) cm (height of the cone), \( r = 6 \) cm, and \( \pi = 3.1416 \).
\[

$$\begin{align*} V_{\text{cone}} &= \frac{1}{3} \times 3.1416 \times 6^2 \times 3 \\ &= \frac{1}{3} \times 3.1416 \times 36 \times 3 \\ &= 3.1416 \times 12 \times 3 \\ &= 3.1416 \times 36 \\ &= 113.0976 \end{align*}$$

\]

Step3: Calculate the volume of the cylinder

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

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

\]

Step4: Calculate the volume of the hemisphere

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

$$\begin{align*} V_{\text{hemisphere}} &= \frac{2}{3} \times 3.1416 \times 6^3 \\ &= \frac{2}{3} \times 3.1416 \times 216 \\ &= 2 \times 3.1416 \times 72 \\ &= 452.3893 \end{align*}$$

\]

Step5: Calculate the total volume

The total volume \( V_{\text{total}} = V_{\text{cone}} + V_{\text{cylinder}} + V_{\text{hemisphere}} \).
\[

$$\begin{align*} V_{\text{total}} &= 113.0976 + 2035.752 + 452.3893 \\ &= 2601.2389 \\ &\approx 2601.24 \end{align*}$$

\]

Answer:

\( V_{\text{cone}} = 113.10 \) \( \text{cm}^3 \) (rounded to two decimal places)
\( V_{\text{cylinder}} = 2035.75 \) \( \text{cm}^3 \) (rounded to two decimal places)
\( V_{\text{hemisphere}} = 452.39 \) \( \text{cm}^3 \) (rounded to two decimal places)
\( V_{\text{total}} = 2601.24 \) \( \text{cm}^3 \) (rounded to two decimal places)