Cylinder Surface Area Ratio Solution

Given Information

The ratio of the curved surface area (CSA) to the total surface area (TSA) of a right circular cylinder is given as $2 : 5$. We need to find the ratio of the height ($h$) to the radius ($r$) of the cylinder.

Formulas Used

• Curved Surface Area (CSA) of a cylinder: $CSA = 2 \pi r h$
• Total Surface Area (TSA) of a cylinder: $TSA = 2 \pi r h + 2 \pi r^2 = 2 \pi r (h + r)$

Calculations

We are given the ratio:

$ \frac{CSA}{TSA} = \frac{2}{5} $

Substitute the formulas for CSA and TSA:

$ \frac{2 \pi r h}{2 \pi r (h + r)} = \frac{2}{5} $

Simplify the expression by cancelling out $2 \pi r$ from the numerator and denominator:

$ \frac{h}{h + r} = \frac{2}{5} $

Now, cross-multiply to solve for the relationship between $h$ and $r$:

$ 5 \times h = 2 \times (h + r) $

$ 5h = 2h + 2r $

Isolate the terms involving $h$ on one side:

$ 5h - 2h = 2r $

$ 3h = 2r $

Finally, find the ratio of the height ($h$) to the radius ($r$) by rearranging the equation:

$ \frac{h}{r} = \frac{2}{3} $

Therefore, the ratio of the height to the radius is $2 : 3$.