Castle model

The castle model has a cone-shaped roof. The cone side is 45 cm long, and the base radius is 27 cm.
a) What is the roof volume?
b) How much dm² of wallpaper is used to glue the roof, i.e., the cone shell?
c) What is the roof's weight if it is made of wood with a density of 0.56 g/cm³?
They were rounded to tenths of a kilogram.

Final Answer:

V =  27482.6525 cm³
S =  38.1704 dm²
m =  15.4 kg

Step-by-step explanation:

$$\begin{gathered} s=45\text{cm} \\ r=27\text{cm} \\ h=\sqrt{s^2-r^2}=\sqrt{45^2-27^2}=36\text{cm} \\ V=\frac{1}{3}\cdot \pi \cdot r^2\cdot h=\frac{1}{3}\cdot 3.1416\cdot 27^2\cdot 36=27482.6525{\text{cm}}^3 \end{gathered}$$

$$\begin{gathered} S_1=\pi \cdot r\cdot s=3.1416\cdot 27\cdot 45\doteq 3817.0351{\text{cm}}^2 \\ S=S_1\to {\text{dm}}^2=S_1:100{\text{dm}}^2=3817.0351:100{\text{dm}}^2=38.17{\text{dm}}^2=38.1704{\text{dm}}^2 \end{gathered}$$

$$\begin{gathered} \rho =0.56{\text{g/cm}}^3 \\ {m}_1=\rho \cdot V=0.56\cdot 27482.6525\doteq 15390.2854\text{g} \\ m=m_1\to \text{kg}=m_1:1000\text{kg}=15390.2854:1000\text{kg}=15.4\text{kg}=15.4\text{kg} \end{gathered}$$

Try another example