Top of the tower

The top of the tower has the shape of a regular hexagonal pyramid. The base edge has a length of 1.2 m, the pyramid height is 1.6 m. How many square meters of sheet metal is needed to cover the top of the tower if 15% extra sheet metal is needed for joints, overlap and waste?

Correct result:

S =  7.8986 m²

Solution:

$$\begin{gathered} a=1.2\text{ m } \\ h=1.6\text{ m } \\ n=6 \\ {h}_1^2+(a\mathrm{/}2{)}^2=a^2 \\ {h}_1=\sqrt{a^2-a^2\mathrm{/}4}=\sqrt{1.2^2-1.2^2\mathrm{/}4}\doteq 1.0392\text{ m } \\ {h}_2=\sqrt{h^2+h_1^2}=\sqrt{1.6^2+1.0392^2}\doteq 1.9079\text{ m } \\ {S}_1=n\cdot {\displaystyle \frac{a\cdot h_2}{2}}=6\cdot {\displaystyle \frac{1.2\cdot 1.9079}{2}}\doteq 6.8684{\text{m}}^2 \\ q=15\mathrm{\%}=1+{\displaystyle \frac{15}{100}}=1.15 \\ S=q\cdot S_1=1.15\cdot 6.8684=7.8986{\text{m}}^2 \end{gathered}$$

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Showing 2 comments:

Mathematican

We need to find the apothem (a) of the base first, so then you can find the height of a triangle of the face since you don't have this, just the height of the pyramid.

0.6² + a² = 1.2² then this a is the apothem, and you can use to find the height of "triangle a face". So h₂ ² = 1.6² + a². And now you do what it is written with these new values.

5 months ago  Like

Mathematican

Thank Luiza. We just corrected this pyramid problem. So h is the height of the whole pyramid, h2 is wall height and h1 is now the height of the base triangles (hexagon is composed of six equilateral triangles).

5 months ago  Like

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