Answer:
Volume = 10,324.6 ft³
Surface area = 2,949 ft²
Step-by-step explanation:
The given figure is a trapezoidal prism with bases that are isosceles trapezoids, as indicated by the congruence of their non-parallel sides.
The volume of a prism is calculated by multiplying the area of one of its parallel bases by its height (the perpendicular distance between the two parallel bases).
[tex]\boxed{\begin{array}{l}\underline{\textsf{Volume of a Prism}}\\\\V=Bh\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$V$ is the volume.}\\\phantom{ww}\bullet\;\textsf{$B$ is the area of the one of the parallel bases.}\\\phantom{ww}\bullet\;\textsf{$h$ is the height of the prism.}\end{array}}[/tex]
To find the volume of the given prism, we first need to find the area of the trapezoidal base.
The formula for the area of a trapezoid is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Area of a trapezoid}}\\\\A=\dfrac{1}{2}h(b_1+b_2)\\\\\textsf{where:}\\ \phantom{ww}\bullet\;\textsf{$A$ is the area.}\\ \phantom{ww}\bullet\;\textsf{$b_1$ and $b_2$ are the parallel sides (bases).}\\\phantom{ww}\bullet\;\textsf{$h$ is the height (perpendicular to the bases).}\end{array}}[/tex]
In this case:
Therefore, the area of base of the prism (B) is:
[tex]B=\dfrac{1}{2} \cdot 24.7(10+34)\\\\\\B=\dfrac{1}{2} \cdot 24.7(44)\\\\\\B=12.35(44)\\\\\\B=543.4\; \sf ft^2[/tex]
The height (h) of the prism is 19 ft. Therefore, substitute B = 543.4 and h = 19 into the volume formula to calculate the volume of the trapezoidal prism:
[tex]V=543.4 \cdot 19\\\\\\V=10324.6\; \sf ft^3[/tex]
Therefore, the volume of the trapezoidal prism is:
[tex]\Large\boxed{\boxed{V=10324.6\; \sf ft^3}}[/tex]
[tex]\dotfill[/tex]
The surface area of a trapezoidal prism is composed of the areas of its two congruent trapezoid bases along with the areas of the four rectangular faces.
We have already determined that the area of one trapezoid base of the prism is 543.5 ft².
The total area of the four rectangular faces can be calculated by multiplying the perimeter of the trapezoid base by the height of the prism:
[tex]\textsf{Area of rectangular faces}=(10+34+27+27) \cdot 19\\\\\\\textsf{Area of rectangular faces}=98 \cdot 19\\\\\\\textsf{Area of rectangular faces}=1862\; \sf ft^2[/tex]
Therefore, the total surface area of the trapezoidal prism is:
[tex]\textsf{Total surface area}=2\;\textsf{Trapezoid base area}+\textsf{Rectangular faces area}\\\\\\\textsf{Total surface area}=2 \cdot 543.5 +1862\\\\\\\textsf{Total surface area}=1087 +1862\\\\\\\textsf{Total surface area}=2949\; \sf ft^2[/tex]
So, the total surface area is:
[tex]\Large\boxed{\boxed{SA=2949\; \sf ft^2}}[/tex]
