find the surface area of the figure to the nearest tenth.

Explanation:

Step1: Calculate lateral area of cone

The formula for the lateral area of a cone is $A_{cone - lateral}=\pi rl$, where $r = 4$ cm and $l=8$ cm. So $A_{cone - lateral}=\pi\times4\times8 = 32\pi$ $cm^{2}$.

Step2: Calculate lateral area of cylinder

The formula for the lateral area of a cylinder is $A_{cylinder - lateral}=2\pi rh$, with $r = 4$ cm and $h = 6$ cm. So $A_{cylinder - lateral}=2\pi\times4\times6=48\pi$ $cm^{2}$.

Step3: Calculate base - area of cylinder

The formula for the area of a circle (base of cylinder) is $A_{base}=\pi r^{2}$, with $r = 4$ cm. So $A_{base}=\pi\times4^{2}=16\pi$ $cm^{2}$.

Step4: Calculate total surface - area

The total surface - area of the figure $A = A_{cone - lateral}+A_{cylinder - lateral}+A_{base}$. Substitute the values: $A=32\pi + 48\pi+16\pi=96\pi$ $cm^{2}$. Using $\pi\approx3.14$, we get $A\approx96\times3.14 = 301.44\approx301.4$ $cm^{2}$.

Answer:

$301.4$ $cm^{2}$