Right triangular prism

A right triangular prism has a base of 6 cm and a base height of h cm. the triangle is a right triangle, and the prism's height is 10 cm. if the total surface area is 288 cm². What is the value of h?

Final Answer:

h =  8

Step-by-step explanation:

$$\begin{gathered} a=6\text{cm} \\ {h}_3=10\text{cm} \\ S=288{\text{cm}}^2 \\ {S}_1=\frac{a\cdot h}{2}=\frac{6\cdot h}{2}=3h \\ {S}_2=(a+h+\sqrt{a^2+h^2})\cdot h_3 \\ S=2\cdot S_1+S_2 \\ S=6h+(a+h+\sqrt{a^2+h^2})\cdot h_3 \\ S=6h+60+10h+10\sqrt{a^2+h^2} \\ (S-60-16h{)}^2=10^2\cdot (a^2+h^2) \\ (288-60-16h{)}^2=10^2\cdot (6^2+h^2) \\ 156h^2-7296h+48384=0 \\ 156=2^2\cdot 3\cdot 13 \\ 7296=2^7\cdot 3\cdot 19 \\ 48384=2^8\cdot 3^3\cdot 7 \\ \text{GCD}(156,7296,48384)=2^2\cdot 3=12 \\ 13h^2-608h+4032=0 \\ a=13;b=-608;c=4032 \\ D=b^2-4ac=608^2-4\cdot 13\cdot 4032=160000 \\ D>0 \\ {h}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{608\pm \sqrt{160000}}{26} \\ {h}_{1,2}=\frac{608\pm 400}{26} \\ {h}_{1,2}=23.384615\pm 15.384615 \\ {h}_1=38.769230769 \\ {h}_2=8 \\ h=h_2=8 \\ \text{ Verifying Solution: } \\ {S}_1=\frac{a\cdot h}{2}=\frac{6\cdot 8}{2}=24{\text{cm}}^2 \\ {S}_2=(a+h+\sqrt{a^2+h^2})\cdot h_3=(6+8+\sqrt{6^2+8^2})\cdot 10=240{\text{cm}}^2 \\ {S}_3=2\cdot S_1+S_2=2\cdot 24+240=288{\text{cm}}^2 \\ S=S_3 \end{gathered}$$

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