Question
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?
To find the volume of the box, we first need to find the volume of the hexagonal prism and the volume of the triangular prism separately, and then add them together.
Volume of the hexagonal prism:
Volume = Base Area x Height
The base of the hexagonal prism is a hexagon with side length of 30 cm.
The area of a regular hexagon = (3√3 x side length^2)/2
Area = (3√3 x 30^2)/2 = (3√3 x 900)/2 = 1350√3 cm^2
Volume = 1350√3 cm^2 x 8 cm = 10800√3 cm^3
Volume of the triangular prism:
Volume = (1/2 x base x height) x height
The base of the triangle is the 15 cm dashed line.
The height of the triangle is given as 8 cm.
Volume = (1/2 x 15 cm x 8 cm) x 8 cm = 480 cm^3
Total volume of the box:
Total volume = Volume of hexagonal prism + Volume of triangular prism
Total volume = 10800√3 cm^3 + 480 cm^3
Total volume ≈ 11133.94 cm^3
Therefore, the volume of the box is approximately 11133.94 cubic centimeters.