To find out the amount of nylon needed to make 50 triangular pyramid (tetrahedron) tea bags, we need to calculate the surface area of one tea bag and then multiply that by 50.
Step 1: Calculate the surface area of one triangular pyramid tea bag.
1. Calculate the area of the triangular base:
The area \(A_{\text{base}}\) of an equilateral triangle can be calculated using the formula: \[ A_{\text{base}} = \frac{\sqrt{3}}{4} s^2 \] where \(s\) is the length of a side of the triangle.
Given the base of the equilateral triangle is 40 mm: \[ A_{\text{base}} = \frac{\sqrt{3}}{4} (40)^2 = \frac{\sqrt{3}}{4} \times 1600 = 400\sqrt{3} \approx 692.82 \text{ mm}^2 \]
2. Calculate the area of the three triangular faces:
Each triangular face has a base equal to the side of the triangular base (40 mm) and a height equal to the slant height (45 mm), so we can calculate the area of one triangular face using: \[ A_{\text{face}} = \frac{1}{2} \times \text{base} \times \text{height} \] Thus, for one face: \[ A_{\text{face}} = \frac{1}{2} \times 40 \times 45 = 900 \text{ mm}^2 \]
Since there are three identical faces: \[ A_{\text{total faces}} = 3 \times 900 = 2700 \text{ mm}^2 \]
3. Calculate the total surface area of one tea bag:
Now we sum the area of the base and the areas of the three triangular faces: \[ A_{\text{total}} = A_{\text{base}} + A_{\text{total faces}} = 400\sqrt{3} + 2700 \approx 692.82 + 2700 = 3392.82 \text{ mm}^2 \]
Step 2: Total area for 50 tea bags
Thus, the total area for 50 tea bags would be: \[ A_{\text{50 tea bags}} = 50 \times A_{\text{total}} \approx 50 \times 3392.82 \approx 169641 \text{ mm}^2 \]
Conclusion:
Looking at the options provided, the closest answer is 170,000 mm².