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.
In a cylinder, the diagonal \( d \) acts as the hypotenuse of a right triangle where one leg is the height \( h \) and the other leg is the diameter of the base, which is \( 2r \).
The relationship can be expressed as:
\[ d^2 = h^2 + (2r)^2 \]
Substituting the known values:
\[ 15^2 = h^2 + (2 \times 5)^2 \]
Calculating \( 15^2 \) and \( (2 \times 5)^2 \):
\[ 225 = h^2 + 100 \]
Now, isolating \( h^2 \):
\[ h^2 = 225 - 100 \] \[ h^2 = 125 \]
Taking the square root of both sides:
\[ h = \sqrt{125} \approx 11.2 \text{ cm} \]
So, the height \( h \) of the cylinder is approximately \( 11.2 \) cm.
Therefore, the answer is:
11.2 cm
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