Question

How many paper cups (of the dimensions shown below) can be made from one A4 paper?

Hint:

For frustum-based problems, include both lateral and base areas if required. Compare the total material available with the required area for clarity.

Answer 1

Step 1: Determine the dimensions of the paper cup.
- The paper cup has a frustum shape with a height of \(6.5 \, \text{cm}\), top diameter \(6.5 \, \text{cm}\), and bottom diameter \(5.5 \, \text{cm}\).
Step 2: Calculate the lateral surface area of one cup.
The lateral surface area of a frustum is given by: \[ \text{Lateral Area} = \pi (r_1 + r_2) \cdot l, \] where \(r_1 = 3.25 \, \text{cm}\), \(r_2 = 2.75 \, \text{cm}\), and \(l = \sqrt{(6.5)^2 + (3.25 - 2.75)^2} \approx 6.52 \, \text{cm}\).
Substituting values: \[ \text{Lateral Area} = \pi (3.25 + 2.75) \cdot 6.52 \approx 125.4 \, \text{cm}^2. \]
Step 3: Compare with the area of an A4 paper.
The total area of A4 paper is \(21 \, \text{cm} \times 29.7 \, \text{cm} = 623.7 \, \text{cm}^2.\)
Step 4: Calculate the number of cups.
Each cup requires \(125.4 \, \text{cm}^2\). The number of cups that can be made is: \[ \text{Number of cups} = \frac{623.7}{125.4} \approx 3. \]