To find the height and the lateral surface area of the cone, we start with the given values: the slant height (\( l \)) is 5 and the diameter is 8, which means the radius (\( r \)) is half of the diameter.
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Calculate the radius: \[ r = \frac{d}{2} = \frac{8}{2} = 4 \]
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Use the Pythagorean theorem to calculate the height (\( h \)) of the cone. The relationship between the radius, height, and slant height is given by: \[ l^2 = r^2 + h^2 \] Plugging in the values we have: \[ 5^2 = 4^2 + h^2 \] \[ 25 = 16 + h^2 \] \[ h^2 = 25 - 16 = 9 \] \[ h = \sqrt{9} = 3 \]
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Now we can calculate the lateral surface area (\( A \)) of the cone using the formula: \[ A = \pi r l \] Substituting \( \pi = 3.14 \), \( r = 4 \), and \( l = 5 \): \[ A = 3.14 \times 4 \times 5 \] \[ A = 3.14 \times 20 = 62.8 \]
So the lateral surface area of the cone is 62.8 square units, and the height of the cone is 3 units.