To find the slant height of the cone, we can use the Pythagorean Theorem. A cone consists of a height, a radius (half of the diameter), and a slant height which acts as the hypotenuse of a right triangle formed by these elements.
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Find the radius: The diameter of the cone is given as 13 cm. Therefore, the radius (r) is: \[ r = \frac{diameter}{2} = \frac{13 \text{ cm}}{2} = 6.5 \text{ cm} \]
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Identify the height: The height (h) of the cone is given as 6 cm.
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Use the Pythagorean Theorem: The relationship between the radius, height, and slant height (l) can be expressed as: \[ l^2 = r^2 + h^2 \]
Substituting the known values: \[ l^2 = (6.5 \text{ cm})^2 + (6 \text{ cm})^2 \] \[ l^2 = 42.25 \text{ cm}^2 + 36 \text{ cm}^2 \] \[ l^2 = 78.25 \text{ cm}^2 \]
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Calculate the slant height: Now take the square root of both sides: \[ l = \sqrt{78.25 \text{ cm}^2} \approx 8.84 \text{ cm} \]
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Round to the nearest tenth: Rounding 8.84 to the nearest tenth gives: \[ l \approx 8.8 \text{ cm} \]
Thus, the required slant height is approximately 8.8 cm.
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