A cylinder of radius $7\text{cm}$ has a curved surface area of $2200\text{cm}{2}^{}$. Find its total surface area. (Use $\pi =\frac{22}{7}$)

The correct answer is 2508 cm².

We are given the following information about a cylinder:

Radius, $r=7\text{cm}$
Curved Surface Area (CSA) $=2200\text{cm}{}^2$
$\pi =\frac{22}{7}$

Step 1: Find the height of the cylinder using the Curved Surface Area formula.

The formula for the Curved (Lateral) Surface Area of a cylinder is:

$\text{CSA}=2\pi rh$

Substituting the known values:

$2200=2\times \frac{22}{7}\times 7\times h$

Simplify $\frac{22}{7}\times 7=22$:

$2200=2\times 22\times h$

$2200=44h$

$h=\frac{2200}{44}=50\text{cm}$

So the height of the cylinder is 50 cm.

Step 2: Find the area of the two circular bases.

The area of one circular base is $\pi r^2$, so the area of two bases is:

$2\pi r^2=2\times \frac{22}{7}\times 7^2$

$=2\times \frac{22}{7}\times 49$

$=2\times 22\times 7$

$=308\text{cm}{}^2$

Step 3: Calculate the Total Surface Area (TSA).

The Total Surface Area of a cylinder is the sum of its Curved Surface Area and the areas of its two circular bases:

$\text{TSA}=\text{CSA}+2\pi r^2$

$\text{TSA}=2200+308$

$\text{TSA}=2508\text{cm}{}^2$

Therefore, the total surface area of the cylinder is 2508 cm².