The ratio of the lateral surface area to the total surface area of a cylinder with base diameter 60cm and height 10 cm is:

The correct option is 1:4.

Given Data:

Base diameter of the cylinder ($d$) = $60\text{ cm}$

Base radius of the cylinder ($r$) = $\frac{60}{2}=30\text{ cm}$

Height of the cylinder ($h$) = $10\text{ cm}$

Formulas:

1. Lateral surface area (Curved surface area) of a cylinder = $2\pi rh$

2. Total surface area of a cylinder = $2\pi r(r+h)$

Step-by-Step Calculation:

We need to find the ratio of the lateral surface area to the total surface area:

$\text{Ratio}=\frac{\text{Lateral Surface Area}}{\text{Total Surface Area}}$

Substitute the formulas into the ratio:

$\text{Ratio}=\frac{2\pi rh}{2\pi r(r+h)}$

Simplifying by cancelling out the common terms $2\pi r$ in the numerator and denominator:

$\text{Ratio}=\frac{h}{r+h}$

Now, substitute the values of $r=30\text{ cm}$ and $h=10\text{ cm}$:

$\text{Ratio}=\frac{10}{30+10}$

$\text{Ratio}=\frac{10}{40}=\frac{1}{4}$

Thus, the ratio of the lateral surface area to the total surface area of the cylinder is 1:4.