An ant walks on the surface of a solid regular hexagonal prism from a corner (A) on the base to the diagonally opposite corner (B) on the top by the shortest route. If the side of the regular hexagon is 21 cm and the height of the prism is 16 cm, then what is the length (in cm) of the path traveled by the ant?
Question
An ant walks on the surface of a solid regular hexagonal prism from a corner (A) on the base to the diagonally opposite corner (B) on the top by the shortest route. If the side of the regular hexagon is 21 cm and the height of the prism is 16 cm, then what is the length (in cm) of the path traveled by the ant?
Solution
To solve this problem, we need to use the Pythagorean theorem.
Step 1: Find the diagonal of the hexagonal base. The hexagon can be divided into six equilateral triangles. The diagonal of the hexagon is twice the side of the equilateral triangle.
The side of the equilateral triangle is 21 cm, so the diagonal of the hexagon is 2*21 = 42 cm.
Step 2: Find the length of the path traveled by the ant. The ant travels from one corner of the base to the diagonally opposite corner of the top. This path forms a right triangle with the diagonal of the base and the height of the prism.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
So, the length of the path traveled by the ant is the square root of (42^2 + 16^2) = square root of (1764 + 256) = square root of 2020 = 44.94 cm (rounded to two decimal places).
So, the length of the path traveled by the ant is approximately 44.94 cm.
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