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
11.5 cm
11.5 cm
14.3 cm
14.3 cm
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To solve this problem, we can use the Pythagorean Theorem.
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.
In this case, the slant height of the cone is the hypotenuse, the height of the cone is one of the sides, and the radius of the base of the cone (which is half of the diameter) is the other side.
We can set up the equation as follows:
$$(\text{slant height})^2 = (\text{height})^2 + (\text{radius})^2$$
Substituting the given values, we have:
$$(\text{slant height})^2 = 6^2 + \left(\frac{13}{2}\right)^2$$
$$(\text{slant height})^2 = 36 + \frac{169}{4}$$
$$(\text{slant height})^2 = \frac{144 + 169}{4}$$
$$(\text{slant height})^2 = \frac{313}{4}$$
Taking the square root of both sides, we have:
$$\text{slant height} = \sqrt{\frac{313}{4}}$$
Approximating to the nearest tenth, we get:
$$\text{slant height} \approx 8.8 \text{ cm}$$
Therefore, the slant height of the cone should be approximately 8.8 cm.
The correct answer is:
8.8 cm