To find the height \( h \) of the cylinder given the radius \( r \) and the diagonal \( d \), we can use the relationship derived from the Pythagorean theorem.
The diagonal \( d \) of the cylinder forms a right triangle with the height \( h \) of the cylinder and the diameter of the base of the cylinder. The diameter \( D \) is twice the radius, so:
\[ D = 2r = 2 \times 5 , \text{cm} = 10 , \text{cm} \]
The formula relating the height, radius, and diagonal for the cylinder is:
\[ d^2 = h^2 + D^2 \]
Substituting in the known values:
\[ 15^2 = h^2 + 10^2 \]
Calculating the squares:
\[ 225 = h^2 + 100 \]
Now, we can isolate \( h^2 \):
\[ h^2 = 225 - 100 \] \[ h^2 = 125 \]
Taking the square root of both sides gives us:
\[ h = \sqrt{125} \approx 11.1803 \]
Rounding to the nearest tenth, we find:
\[ h \approx 11.2 , \text{cm} \]
Thus, the height of the cylinder is:
11.2 cm
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