The Pythagorean Theorem in Three Dimensions Quick Check
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Question
Use the image to answer the question.
A drawing shows a circle showing diameter below a sector of a circle showing slant height. A double-headed arrow crossing the center of the circle is labeled as d equals diameter. A dotted double-headed arrow is drawn from the top point of the sector of the circle to a point along the curved edge. The arrow is labeled as lower L equals slant height.
Chem Tech makes paper cones used to filter liquids in lab experiments. To make the paper cones, the company starts with the net of the shape, where the slant height is needed. The company wants the height of the cones to be 6 cm. If they make the diameter 13 cm, what must the slant height be? Use the Pythagorean Theorem and round the answer to the nearest tenth.
(1 point)
Responses
2.5 cm
2.5 cm
8.8 cm
8.8 cm
14.3 cm
14.3 cm
11.5 cm
To find the slant height, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the diameter of the cone is given as 13 cm, so the radius (which is half the diameter) is 6.5 cm. The height of the cone is given as 6 cm.
We can treat the slant height as the hypotenuse of a right triangle, with one leg being the radius of the base of the cone (6.5 cm) and the other leg being the height of the cone (6 cm).
Using the Pythagorean theorem, we have:
slant height^2 = radius^2 + height^2
slant height^2 = 6.5^2 + 6^2
slant height^2 = 42.25 + 36
slant height^2 = 78.25
Taking the square root of both sides, we find:
slant height ≈ sqrt(78.25)
slant height ≈ 8.8 cm
So the slant height of the cone should be approximately 8.8 cm. Therefore, the correct answer is:
8.8 cm