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 two circular bases (each with area \( \pi r^2 \)) and the lateral (curved) surface area (with area \( 2\pi rh \)).

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

We are given \( r = 14 \, \text{cm} \) and \( h = 30 \, \text{cm} \). First, calculate \( 2\pi r^2 \):

$$ 2\pi r^2 = 2\times\pi\times(14)^2 = 2\times\pi\times196 = 392\pi $$

Next, calculate \( 2\pi rh \):

$$ 2\pi rh = 2\times\pi\times14\times30 = 840\pi $$

Step3: Add the two results together to find the total surface area

$$ A = 392\pi + 840\pi = 1232\pi $$

If we use \( \pi \approx \frac{22}{7} \), then:

$$ A = 1232\times\frac{22}{7} = \frac{1232\times22}{7} = 176\times22 = 3872 \, \text{cm}^2 $$

Answer:

The total area of the unopened can is \( 3872 \, \text{cm}^2 \) (or \( 1232\pi \, \text{cm}^2 \) if leaving it in terms of \( \pi \)).