The bases of a prism are right triangles with side lengths 6 meters,sters, and 10 meters. The height of the prism is 3 meters. What iswho lateral area of the prism? What is the total surface area?

Answer:Part a) The lateral area of the prism is [tex]LA=(48+6\sqrt{34})\ m^2[/tex]Part b) The surface area of the prism is [tex]SA=(108+6\sqrt{34})\ m^2[/tex]Step-by-step explanation:Part a) What is  the lateral area of the prism?we know thatThe lateral area of the prism is[tex]LA=PH[/tex]whereP is the perimeter of the baseH is the height of the prismwe have[tex]a=6\ m\\b=10\ m\\H=3\ m[/tex]The perimeter of the base is[tex]P=a+b+c[/tex]Find the hypotenuse of the right triangleApplying the Pythagoras Theorem[tex]c^2=6^2+10^2[/tex][tex]c^2=136[/tex][tex]c=\sqrt{136}\ m[/tex][tex]c=2\sqrt{34}\ m[/tex]Find the perimeter of the base P[tex]P=6+10+2\sqrt{34}[/tex][tex]P=(16+2\sqrt{34})\ m[/tex]Find the lateral area of the prism[tex]LA=(16+2\sqrt{34})3[/tex][tex]LA=(48+6\sqrt{34})\ m^2[/tex]Part b) What is the total surface area?The total surface area is[tex]SA=LA+2B[/tex]whereLA is the lateral areaB is the area of the baseFind the area of the baseRemember that the base is a triangle so[tex]B=\frac{1}{2}(a)(b)[/tex]we have[tex]a=6\ m\\b=10\ m[/tex]substitute[tex]B=\frac{1}{2}(6)(10)[/tex][tex]B=30\ m^2[/tex]Find the surface area of the prism[tex]SA=LA+2B[/tex]we have[tex]B=30\ m^2[/tex][tex]LA=(48+6\sqrt{34})\ m^2[/tex]substitute[tex]SA=(48+6\sqrt{34})+2(30)[/tex][tex]SA=(48+6\sqrt{34})+60[/tex][tex]SA=(108+6\sqrt{34})\ m^2[/tex]