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We are given that the object is composed of a cone on top of a cylinder. We can use the Pythagorean relationship to find the slant height of the cone.

$\begin{array}{l}{s}^{2}={5}^{2}+{7}^{2}\\ =25+49\\ =74\\ s=\sqrt{74}\end{array}$

$\text{Slant height}=\sqrt{74}$

Then we can use the formula for the surface of a cone, exculding the bottom $\left(S{A}_{cone}=\pi rs\right)$.

For the cylindrical part of the object, we can use the formula for the surface area of a cylinder, but only include one circle, the base $\left({\text{SA}}_{cylinder}=\pi {r}^{2}+2\pi rh\right)$ .

The total surface area can now be calculated:

$\begin{array}{l}{\text{SA}}_{Total}=\pi rs+\pi {r}^{2}+2\pi rh\\ =\pi \left(5\right)\sqrt{74}+\pi {\left(5\right)}^{2}+2\pi \left(5\right)\left(8\right)\\ \approx 464.99\end{array}$

Therefore the surface area is approximately 465 cm ^{ 2 } .

To convert the answer to square inches, use 1 cm = 0.3937 in., so 1 cm^{ 2 } = (0.3937) ^{ 2 } in. ^{ 2 } .

465 (0.3937) ^{ 2 } = 72.074855

Therefore the surface area is approximately 72 in ^{ 2 } .

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