We have to find the length "x".
We know that the area of the prism is 408 cm².
We can express the area of the prism in function of x and then equal this to 408 cm² to find x.
But if we look at the development of the prism, we can see that x is also the hypotenuse of the right triangle:
Then, we can use the Pythagorean theorem to find x and then use the area to check the result.
We can find then the value of x as:
[tex]\begin{gathered} x^2=8^2+6^2 \\ x^2=64+36 \\ x^2=100 \\ x=\sqrt[]{100} \\ x=10 \end{gathered}[/tex]
We now will check with the area of the prism.
The area of the prism will be the sum of the are of all its faces:
[tex]\begin{gathered} A=15\cdot(10+8+6)+\frac{8\cdot6}{2}\cdot2 \\ A=15\cdot24+48 \\ A=360+48 \\ A=408 \end{gathered}[/tex]
Then, as the area we have obtained is 408 cm², the value of x is correct.
Answer: x = 10 cm.
