Question

What is #4 "find the volume and total surface area of the trapezoidal based prism shown At right

Answer 1

so hmmm, notice the picture below

we end up with roughly x = 5.73, y = 4.02 and z = 2.98 rounded up

so, the bases of the trapezoid are, 4 and "x + 4 + z", and its height is "y"

so, the area of the trapezoid is
`\bf \textit{area of a trapezoid}\\\\ A=\cfrac{h}{2}(a+b)\qquad \begin{cases} h=height\\ a,b=\textit{parallel sides, or bases}\\ ----------\\ h=7sin(35^o)\\ a=4\\ b=7cos(35^o)+4+7sin(35^o) \end{cases} \\\\\\ A=\cfrac{4.02}{2}[4+\ (5.73+4+4.02\ )]`

now, the surface area, well, the shape is just two trapezoids, and 3 rectangles stacked up to each other at the edges

so, you know the area of each of the trapezoids already
you have a rectangle of 12*5, another of 7*12, and the third one is 12*(x+4+z)

get the area of the rectangles, sum all areas up, all rectangles and trapezoids, and that's the surface area of the prism

now, the volume, well, is really just the, trapezoidal area, times the length
the length is 12, you already have the area of the trapezoid, their product is the volume