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 answer

To find the volume of the hexagonal prism labeled "Choosy Cheesy," we can use the formula for the volume of a prism, which is given by:

\[ V = B \cdot h \]

where \(B\) is the area of the base, and \(h\) is the height of the prism.

Step 1: Calculate the area of the hexagonal base.

Given that the two parallel sides of the hexagonal prism each measure 30 centimeters in length, it suggests that we are dealing with a regular hexagonal prism. The area \(A\) of a regular hexagon can be calculated using the formula:

\[ A = \frac{3\sqrt{3}}{2} \cdot s^2 \]

where \(s\) is the length of a side.

However, from the problem, we interpret that given the parallel sides are 30 cm, we can find the side length \(s\) of the hexagon. In a regular hexagon, the relationship between the side length and the longer parallel side (which corresponds to the diameter of the hexagon across its width) is \(s \cdot \sqrt{3} = 30\), which means:

\[ s = \frac{30}{\sqrt{3}} = 10\sqrt{3} \text{ centimeters} \]

Next, we calculate the area \(A\) of the hexagon using the derived side length:

\[ A = \frac{3\sqrt{3}}{2} \cdot (10\sqrt{3})^2 \] \[ A = \frac{3\sqrt{3}}{2} \cdot 300 \] \[ A = 450\sqrt{3} \text{ square centimeters} \]

Step 2: Calculate the height of the prism.

The height \(h\) of the hexagonal prism is provided as 8 centimeters.

Step 3: Calculate the volume.

Finally, we can calculate the volume \(V\) of the prism by substituting our values into the volume formula:

\[ V = A \cdot h = (450\sqrt{3}) \cdot 8 \] \[ V = 3600\sqrt{3} \text{ cubic centimeters} \]

Thus, the volume of the box is:

\[ \boxed{3600\sqrt{3}} \text{ cubic centimeters} \]

This volume represents the total capacity of the gourmet cheese box.