Cardboard box

We want to make an open-top cardboard box in the shape of a quadrilateral prism with a rhombus-shaped base. The rhombus has a side of 5 cm and one diagonal of 8 cm. The height of the box is 12 cm. How many square centimetres of cardboard are needed if overlaps and joints require an additional 5%?

Final Answer:

S =  277.2 cm²

Step-by-step explanation:

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

Try another example