To find the height \( h \) of the cylinder given the radius \( r = 5 \) cm and the diagonal \( d = 15 \) cm, we can use the Pythagorean theorem. The diagonal of the cylinder can be thought of as the hypotenuse of a right triangle where one leg is the height \( h \) and the other leg is the diameter of the base of the cylinder.
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First, calculate the diameter of the base: \[ \text{Diameter} = 2r = 2 \times 5 = 10 \text{ cm} \]
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Next, set up the Pythagorean theorem where \( d \) is the hypotenuse, the height \( h \) is one leg, and the diameter is the other leg: \[ d^2 = h^2 + \text{Diameter}^2 \]
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Substituting the known values: \[ 15^2 = h^2 + 10^2 \] \[ 225 = h^2 + 100 \]
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Rearranging the equation to solve for \( h^2 \): \[ h^2 = 225 - 100 \] \[ h^2 = 125 \]
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Taking the square root to find \( h \): \[ h = \sqrt{125} \approx 11.1803 \]
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Rounding to the nearest tenth: \[ h \approx 11.2 \text{ cm} \]
Thus, the height \( h \) of the cylinder is approximately \( \boxed{11.2} \) cm.