Cardboard box

We want to make a cardboard box-shaped quadrangular prism with a rhombic base. The rhombus has a side of 5 cm and 8 cm, one diagonal long. The height of the box is 12 cm. The box will be open at the top. How many square centimeters do we need if we calculate the overlap and joints need 5% of the cardboard?

Correct answer:

S =  277.2 cm²

Step-by-step explanation:

$$\begin{gathered} q=100\%+5\%=1+\frac{5}{100}=1.05 \\ h=12\text{cm} \\ a=5\text{cm} \\ d=8\text{cm} \\ s=\frac{a+a+d}{2}=\frac{5+5+8}{2}=9\text{cm} \\ {S}_1=a\cdot h=5\cdot 12=60{\text{cm}}^2 \\ {S}_2=\sqrt{s\cdot (s-a)\cdot (s-a)\cdot (s-d)}=\sqrt{9\cdot (9-5)\cdot (9-5)\cdot (9-8)}=12{\text{cm}}^2 \\ {S}_3=4\cdot S_1+2\cdot S_2=4\cdot 60+2\cdot 12=264{\text{cm}}^2 \\ S=q\cdot S_3=\frac{21}{20}\cdot 264=\frac{21\cdot 264}{20}=\frac{5544}{20}=\frac{1386}{5}{\text{cm}}^2=277.2{\text{cm}}^2 \end{gathered}$$

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