Rain made an error.
The expression should be
![3\left((1)/(2)*10* \bf{12}\right)+ \left((1)/(2)*10 * 8.7\right)](https://img.qammunity.org/2022/formulas/mathematics/middle-school/2xpmcuq7dnllmucdr88nw4gty6g8w6sof3.png)
Note the term in bold which is 12. I've replaced 13 with this value. This is because the height of each lateral side is 12 inches, instead of 13 inches.
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The steps needed to be taken are therefore:
![3\left((1)/(2)*10* 12\right)+ \left((1)/(2)*10 * 8.7\right)\\\\\\3\left(5* 12\right)+ \left(5 * 8.7\right)\\\\\\3\left(60\right)+ 43.5\\\\\\180+ 43.5\\\\\\223.5\\\\\\](https://img.qammunity.org/2022/formulas/mathematics/middle-school/wtxw296aiarzi0gem9q506fym3ao0nf8hx.png)
This means the triangular pyramid's surface area should be 223.5 square inches.