Question

factory makes nylon tea bags that are in the shape of triangular pyramid how much nylon is needed to make 50 bags
base of equaliateral triangles 40
height of base triangle 35
slant height of equilateral traingle faces is 45

Answer 1

To calculate the amount of nylon needed to make 50 triangular pyramid-shaped tea bags, we first need to calculate the surface area of one triangular pyramid (tetrahedron) and then multiply it by the number of bags.

### Step 1: Calculate the area of the triangular base.

Given, the base is an equilateral triangle with:
- Side length (\( s \)) = 40

The formula for the area \( A \) of an equilateral triangle is:
\[
A = \frac{\sqrt{3}}{4} s^2
\]

Substituting the given side length:
\[
A = \frac{\sqrt{3}}{4} (40)^2 = \frac{\sqrt{3}}{4} \times 1600 = 400\sqrt{3} \approx 692.82 \text{ square units}
\]

### Step 2: Calculate the area of the three triangular faces.

The area of each triangular face can be calculated using the base and height or by using the formula for the area of a triangle:

The area of one triangular face:
- Base = 40 (base of each triangular face)
- Height = 45 (slant height)

Using the area formula for a triangle:
\[
A = \frac{1}{2} \times base \times height
\]
\[
A = \frac{1}{2} \times 40 \times 45 = 900 \text{ square units}
\]

Since there are three triangular faces, the total area of the triangular faces is:
\[
\text{Total Area of triangular faces} = 3 \times 900 = 2700 \text{ square units}
\]

### Step 3: Calculate the total surface area of one triangular pyramid.

Now, we can find the total surface area of one tetrahedron:
\[
\text{Total Surface Area} = \text{Area of base} + \text{Area of 3 triangular faces}
\]
\[
\text{Total Surface Area} = 400\sqrt{3} + 2700
\]
Substituting in the approximation of \(\sqrt{3} \approx 1.732\):
\[
\text{Total Surface Area} \approx 400(1.732) + 2700 = 692.8 + 2700 \approx 3392.8 \text{ square units}
\]

### Step 4: Calculate the total nylon needed for 50 bags.

Finally, to find the total amount of nylon needed for 50 bags:
\[
\text{Total nylon} = 50 \times 3392.8 \approx 169640 \text{ square units}
\]

Thus, the total amount of nylon needed to make 50 equilateral triangular pyramid tea bags is approximately **169640 square units**.