To calculate the total surface area needed to make 50 triangular pyramid-shaped tea bags, we'll need to find the surface area of one triangular pyramid first.
Step 1: Calculate the Area of the Base Triangle
The base of the tea bag is an equilateral triangle. We can calculate its area using the formula:
\[ \text{Area} = \frac{\sqrt{3}}{4} a^2 \]
where \(a\) is the length of a side of the triangle. Given that the base of the triangle is \(40\) mm, we can plug that in:
\[ \text{Area} = \frac{\sqrt{3}}{4} \times (40^2) \]
\[ = \frac{\sqrt{3}}{4} \times 1600 \]
\[ = 400\sqrt{3} , \text{mm}^2 \approx 400 \times 1.732 = 692.8 , \text{mm}^2 \]
Step 2: Calculate the Area of the Triangular Faces
The triangular faces are isosceles triangles. The area of one triangular face is given by the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
For the triangular faces, the base is the same as the side of the triangle \(= 40\) mm, and the height (slant height) is \(45\) mm. Therefore:
\[ \text{Area}_{face} = \frac{1}{2} \times 40 \times 45 \]
\[ = 20 \times 45 = 900 , \text{mm}^2 \]
Step 3: Total Surface Area of One Triangular Pyramid
Now, we have one base and three triangular faces, so the total surface area \(A\) of one tea bag is:
\[ A = \text{Area}{base} + 3 \times \text{Area}{face} \]
\[ A = 692.8 + 3 \times 900 \]
\[ = 692.8 + 2700 \]
\[ = 3392.8 , \text{mm}^2 \]
Step 4: Total Surface Area for 50 Tea Bags
Now, to find the amount of nylon needed for 50 tea bags:
\[ \text{Total Area} = 3392.8 \times 50 \]
\[ = 169640 , \text{mm}^2 \]
Conclusion
Given the nearest round off to significant figures, the total amount of nylon needed is approximately:
170,000 mm²
Therefore, the correct response is:
170,000 mm squared