Question

find the total area of an unopened can of beans with radius r = 14 cm and height h = 30cm.

Explanation:

Step1: Recall the formula for the total surface area of a cylinder

The total surface area \( A \) of a cylinder is given by the formula \( A = 2\pi r^2 + 2\pi rh \), where \( r \) is the radius and \( h \) is the height. This formula accounts for the area of the two circular bases (\( 2\pi r^2 \)) and the lateral (curved) surface area (\( 2\pi rh \)).

Step2: Substitute the given values of \( r \) and \( h \)

We are given \( r = 14 \, \text{cm} \) and \( h = 30 \, \text{cm} \). First, calculate the area of the two bases:
\( 2\pi r^2 = 2 \times \pi \times (14)^2 \)
\( = 2 \times \pi \times 196 \)
\( = 392\pi \)

Next, calculate the lateral surface area:
\( 2\pi rh = 2 \times \pi \times 14 \times 30 \)
\( = 840\pi \)

Step3: Add the two areas to get the total surface area

Now, add the area of the two bases and the lateral surface area:
\( A = 392\pi + 840\pi \)
\( = 1232\pi \)

If we use \( \pi \approx \frac{22}{7} \) for calculation:
\( A = 1232 \times \frac{22}{7} \)
First, divide 1232 by 7: \( 1232 \div 7 = 176 \)
Then, multiply by 22: \( 176 \times 22 = 3872 \)

Answer:

The total surface area of the unopened can is \( 1232\pi \, \text{cm}^2 \) or approximately \( 3872 \, \text{cm}^2 \) (when \( \pi \approx \frac{22}{7} \)).