Answer
• The lateral area of the prism is 900in².
,
• The total area of the prism is 954in².
Explanation
The lateral area of a right triangle prism (AL) is given by:
[tex]A_L=A_{r1}+A_{r2}+A_{r3}[/tex]
where Ar₁ represents the area of the first rectangle, Ar₂ the second, and Ar₃ the third. The area of a rectangle is given by:
[tex]A_r=b\times h[/tex]
We have that b₁ = 12in, b₂ = 15in, and we don't know the value for b₃. However, we can calculate it with the Pythagorean Theorem:
[tex]b_2^2=b_1^2+b_3^2[/tex][tex]b_3^=\sqrt{b_2^2-b_1^2}[/tex][tex]b_3=\sqrt{15^2-12^2}=\sqrt{81}=9[/tex]
Then, the lateral area is the sum of Ar₁ , Ar₂, and Ar₃:
[tex]A_{r1}=12\times25=300[/tex][tex]A_{r2}=15\times25=375[/tex][tex]A_{r3}=9\times25=225[/tex]
Thus, the lateral area is:
[tex]A_L=300+375+225=900in^2[/tex]
Finally, the total area (AT) is the addition of the area of the two bases (Ab) of the prism with the lateral area:
[tex]A_b=\frac{1}{2}\times9\times12=54in^2[/tex][tex]A_T=54+900=954in^2[/tex]
