Q:

# a prism 2m tall. the base is a trapezoid whose parallel sides are 7m and 3m. The other sides are each 4m. The altitude of the trapezoid measures 3.5m. Find the surface area and round it to the nearest tenth, if necessary.

Accepted Solution

A:
To solve this we are going to use the formula for the total surface area of a trapezoidal prism: $$T_{sa}=L_{sa}+2A_{b}$$
where
$$T_{sa}$$ is the total surface area of the trapezoidal prism
$$L_{sa}$$ is the lateral surface area of the trapezoidal prism
$$A_{b}$$ is the Area of base (area of the trapezoid)

To find $$L_{sa}$$, we are going to use the formula for the lateral surface area of a trapezoid: $$L_{sa}=H(a+b+c+d)$$
where
$$a$$ is the larger base of the trapezoid
$$b$$ is the shorter base of the trapezoid
$$c$$ and $$d$$ are the other sides of the trapezoid
Remember that the parallel sides of a trapezoid are its bases, so: $$a=7$$ and $$b=3$$. We also know that the other sides are each 4m, so: $$c=4$$ and $$d=4$$. Also, since the prism is 2m tall, $$H=2$$.  Lets replace those values in our formula to find $$L_{sa}:$$:
$$L_{sa}=H(a+b+c+d)$$
$$L_{sa}=2(7+3+4+4)$$
$$L_{sa}=2(18)$$
$$L_{sa}=36$$$$m^2$$

Now, to find the area of the base, we are going to use the formula for the area of a trapezoid: $$A_{b}=h( \frac{a+b}{2} )$$
where
$$A_{b}$$ is the area of the trapezoid
$$h$$ is the altitude of the trapezoid
$$a$$ is the larger base of the trapezoid
$$b$$ is the shorter base of the trapezoid
We know for our problem that $$h=3.5$$, $$a=7$$, and $$b=3$$. Lets replace those values in our formula to find $$A_{b}$$:
$$A_{b}=h( \frac{a+b}{2} )$$
$$A_{b}=3.5( \frac{7+3}{2} )$$
$$A_{b}=3.5( \frac{10}{2} )$$
$$A_{b}=3.5(5)$$
$$A_{b}=17.5$$$$m^2$$

Now that we have $$L_{sa}$$ and $$A_{b}$$, we can finally use the formula for the total surface area of a trapezoidal prism:
$$T_{sa}=L_{sa}+2A_{b}$$
$$T_{sa}=36+2(17.5)$$
$$T_{sa}=36+35$$
$$T_{sa}=71$$$$m^2$$

We can conclude that the surface area of the prism is 71 square meters.