To solve this we are going to use the formula for the total surface area of a trapezoidal prism: [tex]T_{sa}=L_{sa}+2A_{b}[/tex]
where
[tex]T_{sa}[/tex] is the total surface area of the trapezoidal prism
[tex]L_{sa}[/tex] is the lateral surface area of the trapezoidal prism
[tex]A_{b}[/tex] is the Area of base (area of the trapezoid)
To find [tex]L_{sa}[/tex], we are going to use the formula for the lateral surface area of a trapezoid: [tex]L_{sa}=H(a+b+c+d)[/tex]
where
[tex]a[/tex] is the larger base of the trapezoid
[tex]b[/tex] is the shorter base of the trapezoid
[tex]c[/tex] and [tex]d[/tex] are the other sides of the trapezoid
Remember that the parallel sides of a trapezoid are its bases, so: [tex]a=7[/tex] and [tex]b=3[/tex]. We also know that the other sides are each 4m, so: [tex]c=4[/tex] and [tex]d=4[/tex]. Also, since the prism is 2m tall, [tex]H=2[/tex]. Lets replace those values in our formula to find [tex]L_{sa}:[/tex]:
[tex]L_{sa}=H(a+b+c+d)[/tex]
[tex]L_{sa}=2(7+3+4+4)[/tex]
[tex]L_{sa}=2(18)[/tex]
[tex]L_{sa}=36[/tex][tex]m^2[/tex]
Now, to find the area of the base, we are going to use the formula for the area of a trapezoid: [tex]A_{b}=h( \frac{a+b}{2} )[/tex]
where
[tex]A_{b}[/tex] is the area of the trapezoid
[tex]h[/tex] is the altitude of the trapezoid
[tex]a[/tex] is the larger base of the trapezoid
[tex]b[/tex] is the shorter base of the trapezoid
We know for our problem that [tex]h=3.5[/tex], [tex]a=7[/tex], and [tex]b=3[/tex]. Lets replace those values in our formula to find [tex]A_{b}[/tex]:
[tex]A_{b}=h( \frac{a+b}{2} )[/tex]
[tex]A_{b}=3.5( \frac{7+3}{2} )[/tex]
[tex]A_{b}=3.5( \frac{10}{2} )[/tex]
[tex]A_{b}=3.5(5)[/tex]
[tex]A_{b}=17.5[/tex][tex]m^2[/tex]
Now that we have [tex]L_{sa}[/tex] and [tex]A_{b}[/tex], we can finally use the formula for the total surface area of a trapezoidal prism:
[tex]T_{sa}=L_{sa}+2A_{b}[/tex]
[tex]T_{sa}=36+2(17.5)[/tex]
[tex]T_{sa}=36+35[/tex]
[tex]T_{sa}=71[/tex][tex]m^2[/tex]
We can conclude that the surface area of the prism is 71 square meters.
