The quadrilateral pyramid

The quadrilateral pyramid has a rectangular base of 24 cm x 3.2dm and a body height of 0.4m. Calculate its volume and surface area.

Correct answer:

V =  10.24 dm³
S =  31.3831 dm²

Step-by-step explanation:

$$\begin{gathered} a=24\text{ cm}\to \text{ dm}=24\mathrm{/}10\text{ dm}=2.4\text{ dm } \\ b=3.2\text{ dm } \\ h=0.4\text{ m}\to \text{ dm}=0.4\cdot 10\text{ dm}=4\text{ dm } \\ {S}_1=a\cdot b=2.4\cdot 3.2={\displaystyle \frac{192}{25}}=7.68{\text{dm}}^2 \\ V={\displaystyle \frac{1}{3}}\cdot S_1\cdot h={\displaystyle \frac{1}{3}}\cdot 7.68\cdot 4={\displaystyle \frac{256}{25}}{\text{dm}}^3=10.24{\text{dm}}^3 \end{gathered}$$

$$\begin{gathered} s_2=\sqrt{h^2+(a\mathrm{/}2{)}^2}=\sqrt{4^2+(2.4\mathrm{/}2{)}^2}\doteq 4.1761\text{ dm } \\ {s}_3=\sqrt{h^2+(b\mathrm{/}2{)}^2}=\sqrt{4^2+(3.2\mathrm{/}2{)}^2}\doteq 4.3081\text{ dm } \\ {S}_2={\displaystyle \frac{a\cdot s_3}{2}}={\displaystyle \frac{2.4\cdot 4.3081}{2}}\doteq 5.1698{\text{dm}}^2 \\ {S}_3={\displaystyle \frac{b\cdot s_2}{2}}={\displaystyle \frac{3.2\cdot 4.1761}{2}}\doteq 6.6818{\text{dm}}^2 \\ S=S_1+2\cdot S_2+2\cdot S_3=7.68+2\cdot 5.1698+2\cdot 6.6818=31.3831{\text{dm}}^2 \end{gathered}$$

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