To find the height \( h \) of the cylinder, we can use the relationship between the radius \( r \), the height \( h \), and the diagonal \( d \) of the cylinder. We can apply the Pythagorean theorem, since the height, radius, and diagonal form a right triangle.
The Pythagorean theorem states:
\[ d^2 = r^2 + h^2 \]
We are given:
- \( r = 5 \) cm
- \( d = 15 \) cm
Substituting the values into the formula:
\[ 15^2 = 5^2 + h^2 \] \[ 225 = 25 + h^2 \] \[ h^2 = 225 - 25 \] \[ h^2 = 200 \]
Now, taking the square root of both sides:
\[ h = \sqrt{200} \]
Calculating \( \sqrt{200} \):
\[ h = \sqrt{100 \times 2} = 10\sqrt{2} \approx 10 \times 1.414 \approx 14.14 \text{ cm} \]
Rounding to the nearest tenth:
\[ h \approx 14.1 \text{ cm} \]
Thus, the height \( h \) of the cylinder is approximately 14.1 cm.