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 forms a right triangle with the height and the diameter of the base of the cylinder.
First, we need to find the diameter \( D \) of the cylinder:
\[ D = 2r = 2 \times 5 = 10 \text{ cm} \]
Now, we can apply the Pythagorean theorem to the right triangle formed by the height \( h \), the radius (half of the diameter) \( r \), and the diagonal \( d \):
\[ d^2 = h^2 + r^2 \]
Substituting the values we have:
\[ 15^2 = h^2 + 5^2 \] \[ 225 = h^2 + 25 \]
Now, we solve for \( h^2 \):
\[ h^2 = 225 - 25 \] \[ h^2 = 200 \]
Taking the square root of both sides gives:
\[ h = \sqrt{200} \approx 14.1 \text{ cm} \]
Rounding to the nearest tenth, the height \( h \) is approximately:
\[ h \approx 14.1 \text{ cm} \]
Thus, the correct response is:
14.1 cm