Use the image to answer the question.
An illustration shows a hexagonal prism with the label Choosy Cheesy on the top surface. The height of the hexagonal prism is 8 centimeters. The two longer parallel sides of the hexagonal prism each measure 30 centimeters in length. One end of the prism is marked with a dashed line measuring 15 centimeters across, joining the two opposite vertices of the parallel edges, forming a triangular prism with a perpendicular measure of 8 centimeters.
A sample of gourmet cheeses came in this box. What is the volume of the box?
(1 point)
cm3
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To find the volume of the box, we need to calculate the volume of the hexagonal prism and the volume of the triangular prism separately and then add them together.
Volume of hexagonal prism:
Area of the hexagonal base = (1/2) * apothem length * perimeter.
The apothem length can be calculated using the formula: apothem = (side length / 2) * √3.
apothem = (30 / 2) * √3 = 15√3 cm.
Perimeter of the hexagon = 6 * side length = 6 * 30 = 180 cm.
Area of hexagon = (1/2) * 15√3 * 180 = 1350√3 cm².
Volume of hexagonal prism = 1350√3 * 8 = 10800√3 cm³.
Volume of triangular prism:
Area of the triangle = (1/2) * base * height = (1/2) * 15 * 8 = 60 cm².
Volume of triangular prism = 60 * 8 = 480 cm³.
Total volume of the box = 10800√3 + 480 = 29750.04 cm³.