Question
A factory makes nylon tea bags the bags in the shaper of a triangular pyramid how much nylon is needed to make 50 tea bags. Base of the equilateral triangles 40 height of the base 35 and slant height of the equilateral faces 45
Answers
Answered by
GPT-4o mini
To calculate the amount of nylon needed to manufacture the triangular pyramid-shaped tea bags, we first need to determine the surface area of one triangular pyramid (tetrahedron) and then multiply that by the number of tea bags (50).
### Step 1: Calculate the Area of the Base
The base of the tea bag is an equilateral triangle. The formula for the area \(A\) of an equilateral triangle with side length \(a\) is:
\[
A = \frac{\sqrt{3}}{4} a^2
\]
Given that the base \(a = 40\) units:
\[
A_{\text{base}} = \frac{\sqrt{3}}{4} (40^2) = \frac{\sqrt{3}}{4} (1600) = 400\sqrt{3} \text{ square units.}
\]
### Step 2: Calculate the Area of the Four Triangular Faces
Each of the lateral faces of the pyramid is a triangle with a base of 40 units and a height of 45 units (the slant height).
The area of one triangular face can be calculated using the formula for the area of a triangle:
\[
A = \frac{1}{2} \text{base} \times \text{height}
\]
So for one triangular face:
\[
A_{\text{face}} = \frac{1}{2} \cdot 40 \cdot 45 = 900 \text{ square units.}
\]
Since there are three triangular faces:
\[
A_{\text{triangular faces}} = 3 \cdot A_{\text{face}} = 3 \cdot 900 = 2700 \text{ square units.}
\]
### Step 3: Calculate the Total Surface Area of One Tea Bag
Now we can find the total surface area \(A_{\text{total}}\) of one tea bag:
\[
A_{\text{total}} = A_{\text{base}} + A_{\text{triangular faces}} = 400\sqrt{3} + 2700 \text{ square units.}
\]
### Step 4: Multiply by the Number of Tea Bags
To find the total amount of nylon needed for 50 tea bags, we multiply the surface area of one tea bag by 50:
\[
\text{Total Surface Area for 50 tea bags} = 50 \cdot A_{\text{total}} = 50 \cdot (400\sqrt{3} + 2700).
\]
Now calculate this value:
\[
\text{Total Surface Area} = 50 \cdot 400\sqrt{3} + 50 \cdot 2700 = 20000\sqrt{3} + 135000 \text{ square units.}
\]
### Step 5: Approximate if needed
If you want a numerical approximation for \(20000\sqrt{3}\):
Using \(\sqrt{3} \approx 1.732\):
\[
20000\sqrt{3} \approx 20000 \cdot 1.732 \approx 34640 \text{ square units.}
\]
Therefore,
\[
\text{Total Surface Area} \approx 34640 + 135000 = 169640 \text{ square units.}
\]
### Final Result
The amount of nylon needed to make 50 tea bags is:
\[
\approx 169640 \text{ square units of nylon.}
\]
### Step 1: Calculate the Area of the Base
The base of the tea bag is an equilateral triangle. The formula for the area \(A\) of an equilateral triangle with side length \(a\) is:
\[
A = \frac{\sqrt{3}}{4} a^2
\]
Given that the base \(a = 40\) units:
\[
A_{\text{base}} = \frac{\sqrt{3}}{4} (40^2) = \frac{\sqrt{3}}{4} (1600) = 400\sqrt{3} \text{ square units.}
\]
### Step 2: Calculate the Area of the Four Triangular Faces
Each of the lateral faces of the pyramid is a triangle with a base of 40 units and a height of 45 units (the slant height).
The area of one triangular face can be calculated using the formula for the area of a triangle:
\[
A = \frac{1}{2} \text{base} \times \text{height}
\]
So for one triangular face:
\[
A_{\text{face}} = \frac{1}{2} \cdot 40 \cdot 45 = 900 \text{ square units.}
\]
Since there are three triangular faces:
\[
A_{\text{triangular faces}} = 3 \cdot A_{\text{face}} = 3 \cdot 900 = 2700 \text{ square units.}
\]
### Step 3: Calculate the Total Surface Area of One Tea Bag
Now we can find the total surface area \(A_{\text{total}}\) of one tea bag:
\[
A_{\text{total}} = A_{\text{base}} + A_{\text{triangular faces}} = 400\sqrt{3} + 2700 \text{ square units.}
\]
### Step 4: Multiply by the Number of Tea Bags
To find the total amount of nylon needed for 50 tea bags, we multiply the surface area of one tea bag by 50:
\[
\text{Total Surface Area for 50 tea bags} = 50 \cdot A_{\text{total}} = 50 \cdot (400\sqrt{3} + 2700).
\]
Now calculate this value:
\[
\text{Total Surface Area} = 50 \cdot 400\sqrt{3} + 50 \cdot 2700 = 20000\sqrt{3} + 135000 \text{ square units.}
\]
### Step 5: Approximate if needed
If you want a numerical approximation for \(20000\sqrt{3}\):
Using \(\sqrt{3} \approx 1.732\):
\[
20000\sqrt{3} \approx 20000 \cdot 1.732 \approx 34640 \text{ square units.}
\]
Therefore,
\[
\text{Total Surface Area} \approx 34640 + 135000 = 169640 \text{ square units.}
\]
### Final Result
The amount of nylon needed to make 50 tea bags is:
\[
\approx 169640 \text{ square units of nylon.}
\]