Prism

A right-angled prism, whose base is a right triangle with leg a = 3 cm and hypotenuse c = 6 cm, has the same volume as a cube with an edge length of 1 dm.

a) Find the height of the prism
b) Calculate the surface of the prism
c) What percentage of the cube's surface is prism surface area?

Final Answer:

h =  128.3 cm
S =  1837 cm²
x =  306.2 %

Step-by-step explanation:

$$\begin{gathered} a=3\text{cm} \\ c=6\text{cm} \\ k=1\text{dm}\to \text{cm}=1\cdot 10\text{cm}=10\text{cm} \\ {c}^2=a^2+b^2 \\ b=\sqrt{c^2-a^2}=\sqrt{6^2-3^2}=3\sqrt{3}\text{cm}\doteq 5.1962\text{cm} \\ {S}_1=\frac{a\cdot b}{2}=\frac{3\cdot 5.1962}{2}\doteq 7.7942{\text{cm}}^2 \\ V=k^3=10^3=1000{\text{cm}}^3 \\ V=S_1\cdot h \\ h=V/S_1=1000/7.7942=128.3\text{cm} \end{gathered}$$

$$\begin{gathered} o=a+b+c=3+5.1962+6\doteq 14.1962\text{cm} \\ S=2\cdot S_1+o\cdot h=2\cdot 7.7942+14.1962\cdot 128.3001=1837{\text{cm}}^2 \end{gathered}$$

$$\begin{gathered} S_2=6\cdot k^2=6\cdot 10^2=600{\text{cm}}^2 \\ x=100\cdot \frac{S}{S_2}=100\cdot \frac{1836.9557}{600}=306.2\% \end{gathered}$$

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