Bot GPT 4-o mini To find the height \( h \) of the cylinder, we can use the relationship between the radius \( r \), height \( h \), and the diagonal \( d \) of the cylinder. The diagonal \( d \) forms the hypotenuse of a right triangle, where the radius \( r \) is one leg, and the height \( h \) is the other leg.
According to the Pythagorean theorem:
\[ d^2 = r^2 + h^2 \]
Given:
- \( r = 5 \) cm
- \( d = 15 \) cm
We can substitute these values into the equation:
\[ 15^2 = 5^2 + h^2 \]
Calculating the squares:
\[ 225 = 25 + h^2 \]
Now, solve for \( h^2 \):
\[ h^2 = 225 - 25 \] \[ h^2 = 200 \]
Now, take the square root of both sides to find \( h \):
\[ h = \sqrt{200} \]
Calculating the square root:
\[ h \approx 14.14 \]
Rounding to the nearest tenth, we find:
\[ h \approx 14.1 \text{ cm} \]
Thus, the height \( h \) of the cylinder is \( \boxed{14.1} \) cm.
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