The radius of the base of a closed right circular cylinder is 21 cm and its height is 1 m. Find its total surface area.

Question

Accepted Answer

The correct option is 3:
To solve this problem, we need to find the total surface area of a closed right circular cylinder. We will recall the formula for the total surface area of a cylinder and then substitute the given values.

First, let's identify the given information:
Radius of the base ($r$) = 21 cm
Height of the cylinder ($h$) = 1 m

Before we proceed with the calculation, it's crucial to ensure that all units are consistent. The radius is given in centimeters (cm), and the height is given in meters (m). We need to convert the height from meters to centimeters.
We know that 1 m = 100 cm.
So, $h = 1 \text{ m} = 1 \times 100 \text{ cm} = 100 \text{ cm}$.

Now, let's recall the formula for the total surface area (TSA) of a closed right circular cylinder. The total surface area of a closed cylinder includes the area of the two circular bases and the curved surface area.
The formula for the total surface area of a cylinder is:
$$TSA = 2\pi r(r + h)$$
Where:
$r$ is the radius of the base
$h$ is the height of the cylinder
$\pi$ is a mathematical constant, approximately $\frac{22}{7}$ or 3.14159. For calculations involving multiples of 7, using $\frac{22}{7}$ is often more convenient.

Now, let's substitute the values of $r = 21 \text{ cm}$ and $h = 100 \text{ cm}$ into the formula:
$$TSA = 2 \times \frac{22}{7} \times 21 \text{ cm} \times (21 \text{ cm} + 100 \text{ cm})$$
$$TSA = 2 \times \frac{22}{7} \times 21 \text{ cm} \times (121 \text{ cm})$$

We can simplify the expression by canceling out the 7 in the denominator with 21:
$$TSA = 2 \times 22 \times 3 \text{ cm} \times 121 \text{ cm}$$
$$TSA = 44 \times 3 \text{ cm} \times 121 \text{ cm}$$
$$TSA = 132 \text{ cm} \times 121 \text{ cm}$$

Now, let's perform the multiplication:
$132 \times 121$
$132 \times (100 + 20 + 1) = 132 \times 100 + 132 \times 20 + 132 \times 1$
$= 13200 + 2640 + 132$
$= 15840 + 132$
$= 15972$

So, the total surface area of the cylinder is $15972 	ext{ cm}^2$.

The correct option is Option 3, which is $15972	ext{ cm}^2$. This matches our calculated value.

The incorrect option is 1: $2409\text{ cm}^2$
This value is significantly smaller than the correct total surface area. A common mistake leading to a much smaller value could be an error in unit conversion (e.g., using 1 cm for height instead of 100 cm) or an incorrect formula calculation, possibly missing one of the terms or using only the area of one base. If one were to calculate only the area of the two bases ($2\pi r^2 = 2 \times \frac{22}{7} \times 21^2 = 2 \times \frac{22}{7} \times 441 = 2 \times 22 \times 63 = 2772 \text{ cm}^2$), this is still higher than 2409, so this option is likely due to a miscalculation or misunderstanding of the formula components.

The incorrect option is 2: $15297\text{ cm}^2$
This value is close to the correct answer but slightly smaller. This discrepancy might arise from using an approximated value for $\pi$ (like 3.14) instead of $\frac{22}{7}$ during intermediate steps, or a minor calculation error during the multiplication. For example, if there was a small error in the sum (r+h) or multiplication.

The incorrect option is 4: $2904\text{ cm}^2$
This value is also significantly smaller than the correct answer. This could result from a calculation error where only the curved surface area ($2\pi rh$) was calculated with an incorrect height, or if the height was taken as 1 cm instead of 100 cm. For instance, if $h=1$ cm: $TSA = 2 \times \frac{22}{7} \times 21 \times (21 + 1) = 2 \times 22 \times 3 \times 22 = 132 \times 22 = 2904 \text{ cm}^2$. This precisely matches option 4, indicating a critical error in unit conversion by not converting 1 m to 100 cm.