Grace has a small business that sells cooler bags filled with cold drinks to tourists

going on game drives. She sells a variety of drinks all packaged in the same type of

cold drink can.

The cold drink cans are 111,5 mm high and have a diameter of 6 cm.


Show that the surface area of the can is 26 675,58 mm2.

𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟 = (2𝜋𝔯 × ℎ) + 2𝜋𝑟2


Let's solve this step by step using the formula provided:

The formula for the surface area of a cylinder is:

\[
S = (2\pi r h) + 2\pi r^2
\]

Where:
- \( r \) is the radius of the cylinder (half the diameter),
- \( h \) is the height of the cylinder,
- \( \pi \) is a constant approximately equal to 3.1416.

### Step 1: Calculate the Radius
The diameter of the cold drink can is 6 cm. To find the radius:

\[
r = \frac{6 \, \text{cm}}{2} = 3 \, \text{cm} = 30 \, \text{mm}
\]

### Step 2: Use Given Height
The height \( h \) of the can is given as 111.5 mm.

### Step 3: Calculate the Surface Area
Now, plug the values into the formula.

- \( r = 30 \, \text{mm} \)
- \( h = 111.5 \, \text{mm} \)

\[
S = (2\pi \times 30 \times 111.5) + 2\pi \times (30)^2
\]

First, calculate the lateral surface area \( (2\pi r h) \):

\[
2\pi \times 30 \times 111.5 = 21027.72 \, \text{mm}^2
\]

Then, calculate the area of the two circular bases \( 2\pi r^2 \):

\[
2\pi \times 30^2 = 5657.86 \, \text{mm}^2
\]

### Step 4: Sum the Two Areas
Now, add the lateral surface area and the area of the circular bases:

\[
S = 21027.72 + 5657.86 = 26,685.58 \, \text{mm}^2
\]

So, the surface area of the can is approximately **26,675.58 mm²**.

This confirms that the surface area of the can is as stated.

Would you like more details or explanations on any step?

### Relative Questions:
1. How would the surface area change if the diameter of the can increased by 2 cm?
2. How does the surface area formula relate to the volume formula of a cylinder?
3. If the height of the can was doubled, how would that affect the surface area?
4. Can you derive the surface area formula for a cylinder starting from first principles?
5. How would the formula change if the can had a conical top instead of being cylindrical?

### Tip:
Always ensure to use consistent units when performing calculations to avoid errors.

Grace has a small business that sells cooler bags filled with cold drinks to tourists going on game drives. She sells a variety of drinks all packaged in the same type of cold drink can. The cold drink cans are 111,5 mm high and have a diameter of 6 cm. Show that the surface area of the can is 26 675,58 mm2.

Discover how to calculate the surface area of a cold drink can with a height of 111.5 mm and a diameter of 6 cm. Detailed step-by-step solution using the formula for the surface area of cylinders. Suitable for high school students studying geometry and surface area calculations.

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