# 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 many dm2 of wallpaper is used to glue the roof, ie the cone shell?
c) What is the weight of the roof if it is made of wood with a density of 0.56 g/cm3?
They were rounded to tenths of a kilogram.

Result

V =  27482.653 cm3
S =  38.17 dm2
m =  15.4 kg

#### Solution:

$s=45 \ \text{cm} \ \\ r=27 \ \text{cm} \ \\ \ \\ h=\sqrt{ s^2 - r^2 }=\sqrt{ 45^2 - 27^2 }=36 \ \text{cm} \ \\ \ \\ V=\dfrac{ 1 }{ 3 } \cdot \ \pi \cdot \ r^2 \cdot \ h=\dfrac{ 1 }{ 3 } \cdot \ 3.1416 \cdot \ 27^2 \cdot \ 36 \doteq 27482.6525 \doteq 27482.653 \ \text{cm}^3$
$S_{1}=\pi \cdot \ r \cdot \ s=3.1416 \cdot \ 27 \cdot \ 45 \doteq 3817.0351 \ \text{cm}^2 \ \\ S=S_{1} \rightarrow dm^2=S_{1} / 100 \ dm^2=38.17035 \ dm^2=38.17 \ \text{dm}^2$
$ρ=0.56 \ \text{g/cm}^3 \ \\ m_{1}=ρ \cdot \ V=0.56 \cdot \ 27482.6525 \doteq 15390.2857 \ \text{g} \ \\ \ \\ m=m_{1} \rightarrow kg=m_{1} / 1000 \ kg=15.39029 \ kg=15.4 \ \text{kg}$

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