Triangle greenhouse

A greenhouse has the shape of a prism lying on its side wall. The base consists of a trapezoid and a triangle. The lower base of the trapezoid has a length of 3 m, the upper base (and the side of the triangle) has a length of 2 m, the height of the trapezoid is 1.8 m, and the height of the triangle is 0.6 m. The height of the prism is 5 m. Calculate:
a) how many m² of glass are needed for its manufacture if 10% of the surface consists of metal structure.
b) What is the volume of the greenhouse.

Final Answer:

S =  27.3091 m²
V =  154.7172 m³

Step-by-step explanation:

$$\begin{gathered} a=3\text{m} \\ c=2\text{m} \\ {h}_1=1.8\text{m} \\ {h}_2=0.6\text{m} \\ l=5\text{m} \\ q=100\%-10\%=1-\frac{10}{100}=0.9 \\ x=(a-c)/2=(3-2)/2=\frac{1}{2}=0.5\text{m} \\ b=\sqrt{h_1^2+x^2}=\sqrt{1.8^2+0.5^2}\doteq 1.8682\text{m} \\ r=\sqrt{h_2^2+(c/2{)}^2}=\sqrt{0.6^2+(2/2{)}^2}\doteq 1.1662\text{m} \\ o=2\cdot b+2\cdot r=2\cdot 1.8682+2\cdot 1.1662\doteq 6.0687\text{m} \\ {S}_1=o\cdot l=6.0687\cdot 5\doteq 30.3434{\text{m}}^2 \\ S=q\cdot S_1=0.9\cdot 30.3434=27.3091{\text{m}}^2 \end{gathered}$$

$$\begin{gathered} S_2=\frac{a+c}{2}\cdot h_1=\frac{3+2}{2}\cdot 1.8=\frac{3}{5}=0.6{\text{m}}^2 \\ {S}_2=\frac{c\cdot h_2}{2}=\frac{2\cdot 0.6}{2}=\frac{3}{5}=0.6{\text{m}}^2 \\ {S}_3=S_1+S_2=30.3434+0.6\doteq 30.9434{\text{m}}^2 \\ V=S_3\cdot l=30.9434\cdot 5=154.7172{\text{m}}^3 \end{gathered}$$

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