The plaster cast

The plaster cast has the shape of a regular quadrilateral pyramid. The cover consists of four equilateral triangles with a 5 m side. Calculate its volume and surface area.

Correct answer:

V =  29.4628 m³
S =  68.3013 m²

Step-by-step explanation:

$$\begin{gathered} a=5\text{m} \\ {S}_1=a^2=5^2=25{\text{m}}^2 \\ {a}^2=(a/2{)}^2+h_1^2 \\ {h}_1=\sqrt{a^2-(a/2{)}^2}=\sqrt{5^2-(5/2{)}^2}\doteq 4.3301\text{m} \\ {h}_1^2=h_2^2+(a/2{)}^2 \\ {h}_2=\sqrt{h_1^2-(a/2{)}^2}=\sqrt{4.3301^2-(5/2{)}^2}\doteq 3.5355\text{m} \\ V=\frac{1}{3}\cdot S_1\cdot h_2=\frac{1}{3}\cdot 25\cdot 3.5355=29.4628{\text{m}}^3 \end{gathered}$$

$$\begin{gathered} S_2=\frac{1}{2}\cdot a\cdot h_1=\frac{1}{2}\cdot 5\cdot 4.3301\doteq 10.8253{\text{m}}^2 \\ S=4\cdot S_2+S_1=4\cdot 10.8253+25=68.3013{\text{m}}^2 \end{gathered}$$

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