Use the image to answer the question.
An illustration shows a table in the shape of a 3-dimensional hexagonal prism, and its 2-dimensional unfolded version.
The 3-dimensional prism shows a hexagon tabletop and table bottom with six vertical rectangular sides connecting the six edges of each hexagon to the other, and also connected to each other. The unfolded version shows 6 vertical rectangles connected to one another horizontally with a vertical length of 75 centimeters. Two identical hexagons are adjoined to the top and bottom of the first vertical rectangle. The diagonal of the top hexagon, shown as a dotted line, is labeled as 90 centimeters. A side of the bottom hexagon is labeled as 45 centimeters, and the perpendicular height from the center of the hexagon to the middle of a side is labeled as 38.97 centimeters. The perpendicular height is denoted by a right angle symbol.
First Choice Interior makes a coffee table in the shape of a regular hexagonal prism. The top is made of wood, the sides of glass, and the bottom of metal. Use the net to find how many cm2 of glass are needed for the furniture company to make one coffee table.
(1 point)
Responses
20,250 cm2
20,250 cm squared
30,772 cm2
30,772 cm squared
17,550 cm2
To find the total surface area of the hexagonal prism, we need to find the area of each individual shape on the net and add them up.
Area of the hexagon:
The formula for the area of a regular hexagon is (3√3/2) * s^2, where s is the length of a side.
Given that the diagonal of the top hexagon is 90 cm, and using the Pythagorean theorem, we can calculate the side length as s = 90/sqrt(3) = 51.96152423 cm.
So, the area of one hexagon is (3√3/2) * (51.96152423)^2 = 6549.52 cm^2.
Since there are two hexagons, the total area of the top and bottom is 2 * 6549.52 = 13099.04 cm^2.
Area of the 6 vertical rectangles:
The area of each rectangle is given by 75 * 38.97 = 2923.25 cm^2.
Since there are 6 rectangles, the total area is 6 * 2923.25 = 17539.5 cm^2.
Therefore, the total surface area of the hexagonal prism is 13099.04 (top and bottom) + 17539.5 (sides) = 30638.54 cm^2.
Since the sides of the coffee table are made of glass, the company would need 17539.5 cm^2 of glass to make one coffee table.
Therefore, the correct answer is 17,550 cm2.