To find the height \( h \) of the cylinder using the given radius \( r = 5 \) cm and the diagonal \( d = 15 \) cm, we can use the Pythagorean theorem.
In a cylinder, when you draw a diagonal \( d \) from the top edge to the opposite bottom edge while having a vertical height \( h \) and the radius \( r \) as the horizontal distance from the center to the edge, you form a right triangle. The relationship can be expressed as:
\[ d^2 = h^2 + r^2 \]
Given:
- \( r = 5 \) cm
- \( d = 15 \) cm
Substituting these values into the equation:
\[ 15^2 = h^2 + 5^2 \]
Calculating the squares:
\[ 225 = h^2 + 25 \]
Now, solve for \( h^2 \):
\[ h^2 = 225 - 25 \] \[ h^2 = 200 \]
Taking the square root of both sides to find \( h \):
\[ h = \sqrt{200} = \sqrt{100 \cdot 2} = 10\sqrt{2} \approx 14.14 \]
Rounding to the nearest tenth, we get:
\[ h \approx 14.1 \text{ cm} \]
Thus, the height \( h \) of the cylinder is:
14.1 cm