A manufacturer uses a mold to make a part in the shape of a triangular prism. The dimensions of this part are shown below.
A triangular prism is shown. The height of the prism is 8.25 millimeters. The sides of the triangular base measure 7.5 , 7.5 , and 14.4 millimeters. A line segment extends from the 14.4 millimeter side to the opposite vertex. The segment is labeled 2.1 millimeters. A small square is at the intersection of this segment with the side of the triangle.
Which estimate is closest to the volume in cubic millimeters of the part?
A. 65
B. 125
C. 249
D. 446
The triangular prism can be divided into two smaller prisms: a larger one with base 7.5 mm by 14.4 mm and height 8.25 mm, and a smaller one that is a pyramid with base 2.1 mm by 2.1 mm and height 8.25 mm.
The volume of the larger prism is $$\frac{1}{2}(7.5\text{ mm})(14.4\text{ mm})(8.25\text{ mm}) = 682.5\text{ mm}^3.$$
The volume of the smaller pyramid is $$\frac{1}{3}(2.1\text{ mm})(2.1\text{ mm})(8.25\text{ mm}) \approx 9.58\text{ mm}^3.$$
Therefore, the total volume is approximately $682.5+9.58\approx \boxed{\textbf{(D)}\ 446}$ cubic millimeters.