A cylinder has a surface area (including ends) of 180.6 sq cm and a diameter of 5 cm. What is the height?

To find the height of the cylinder given its surface area and diameter, we begin by recalling the formula for the total surface area (A) of a cylinder:

[ A = 2\pi r^2 + 2\pi rh ]

Here, ( r ) is the radius, ( h ) is the height, and ( \pi ) is a constant approximately equal to 3.14. Given a diameter of 5 cm, we can calculate the radius:

[ r = \frac{diameter}{2} = \frac{5 , cm}{2} = 2.5 , cm ]

Now we substitute the known values into the surface area formula. We are given the total surface area being 180.6 sq cm:

[ 180.6 = 2\pi (2.5^2) + 2\pi (2.5)h ]

Calculating the area of the circular ends:

[ 2\pi (2.5^2) = 2\pi (6.25) = 12.5\pi \approx 39.27 , sq , cm ]

Now we substitute this into our equation:

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