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
To find the volume of the hexagonal prism, we need to find the area of the hexagonal base and then multiply it by the height of the prism.
Since the hexagon is regular, we can use the formula for the area of a regular hexagon:
Area of a regular hexagon = 3√3 * s², where s is the length of the side.
In this case, the side length of the hexagon is 30 cm. So, the area of the hexagonal base is:
Area = 3√3 * (30)²
Area = 3√3 * 900
Area = 2700√3 cm²
Now, to find the volume of the hexagonal prism, we multiply the area of the base by the height:
Volume = 2700√3 * 8
Volume = 21600√3 cubic cm
Therefore, the volume of the box is 21600√3 cubic cm.
Wrong, please recalculate...
Apologies for the mistake. Let's recalculate.
To find the volume of the box, we first need to find the volume of the hexagonal prism. The formula for the volume of a prism is V = base area x height.
First, let's find the area of the hexagonal base. Since the hexagon is regular, we can divide it into six congruent equilateral triangles. The formula for the area of an equilateral triangle is A = (sqrt(3) / 4) x side length^2.
Given that the side length of the hexagon is 30 cm, the area of one equilateral triangle is:
A = (sqrt(3) / 4) x (30)^2
A = (sqrt(3) / 4) x 900
A = 225(sqrt(3)) cm^2
Since there are six equilateral triangles in the hexagon, the total area of the hexagonal base is:
Total Area = 6 x 225(sqrt(3))
Total Area = 1350(sqrt(3)) cm^2
The height of the hexagonal prism is 8 cm.
Now, we can calculate the volume of the hexagonal prism:
V = Total Area x height
V = 1350(sqrt(3)) x 8
V = 10800(sqrt(3)) cm^3
Therefore, the volume of the box is 10800(sqrt(3)) cubic cm.