Question

what is the surface area of this figure?
round your answer to the nearest hundredth.
square feet

Explanation:

Step1: Calculate the surface area of the rectangular - prism part

The formula for the surface area of a rectangular prism is \(S_{prism}=2(lw + lh+wh)\). But since the top of the rectangular prism is covered by the half - cylinder, we need to adjust.
The area of the base is \(lw = 12\times4=48\) square feet.
The area of the two side faces: \(2\times(wh)=2\times(4\times6) = 48\) square feet.
The area of the two back - and - front non - covered faces: \(2\times(lh)=2\times(12\times6)=144\) square feet.

Step2: Calculate the surface area of the half - cylinder part

The formula for the lateral surface area of a full cylinder is \(S_{lateral}=2\pi rh\), for a half - cylinder, \(S_{half - lateral}=\pi rh\). Here, the diameter of the cylinder is equal to the width of the rectangular prism, so \(r=\frac{4}{2}=2\) feet and \(h = 12\) feet. Then \(S_{half - lateral}=\pi\times2\times12=24\pi\) square feet.
The area of the two semi - circles (which together form a full circle) is \(S_{circle}=\pi r^{2}=\pi\times2^{2}=4\pi\) square feet.

Step3: Sum up all the areas

\(S=48 + 48+144+24\pi+4\pi\)
\(S=240 + 28\pi\)
Substitute \(\pi\approx3.14159\)
\(S=240+28\times3.14159\)
\(S=240 + 87.96452\)
\(S=327.96452\approx327.96\)

Answer:

327.96