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To find the lower and upper bounds of the perimeter of an equilateral triangle with a given area of [tex]\( 100 \, cm^2 \)[/tex] measured to the nearest [tex]\( 10 \, cm^2 \)[/tex], follow these steps:
### Step 1: Determine the range for the area
Given that the area of the triangle is measured to the nearest [tex]\( 10 \, cm^2 \)[/tex], the actual area can vary within a range of [tex]\( \pm 5 \, cm^2 \)[/tex] from 100 [tex]\( cm^2 \)[/tex]. Therefore:
- The lower bound for the area is [tex]\( 100 - 5 = 95 \, cm^2 \)[/tex]
- The upper bound for the area is [tex]\( 100 + 5 = 105 \, cm^2 \)[/tex]
### Step 2: Recall the formula for the area of an equilateral triangle
The area [tex]\( A \)[/tex] of an equilateral triangle with side length [tex]\( s \)[/tex] is given by:
[tex]\[ A = \frac{\sqrt{3}}{4} s^2 \][/tex]
### Step 3: Find the side lengths for the lower and upper bounds of the area
- For the lower bound area of [tex]\( 95 \, cm^2 \)[/tex]:
[tex]\[ 95 = \frac{\sqrt{3}}{4} s_{\text{lower}}^2 \][/tex]
[tex]\[ s_{\text{lower}}^2 = \frac{95 \times 4}{\sqrt{3}} \][/tex]
[tex]\[ s_{\text{lower}} = \sqrt{\frac{380}{\sqrt{3}}} \][/tex]
[tex]\[ s_{\text{lower}} \approx 14.8119 \, cm \][/tex]
- For the upper bound area of [tex]\( 105 \, cm^2 \)[/tex]:
[tex]\[ 105 = \frac{\sqrt{3}}{4} s_{\text{upper}}^2 \][/tex]
[tex]\[ s_{\text{upper}}^2 = \frac{105 \times 4}{\sqrt{3}} \][/tex]
[tex]\[ s_{\text{upper}} = \sqrt{\frac{420}{\sqrt{3}}} \][/tex]
[tex]\[ s_{\text{upper}} \approx 15.5720 \, cm \][/tex]
### Step 4: Calculate the perimeter for the lower and upper bounds
The perimeter [tex]\( P \)[/tex] of an equilateral triangle with side length [tex]\( s \)[/tex] is given by:
[tex]\[ P = 3s \][/tex]
- For the lower bound side length [tex]\( 14.8119 \, cm \)[/tex]:
[tex]\[ P_{\text{lower}} = 3 \times 14.8119 \][/tex]
[tex]\[ P_{\text{lower}} \approx 44.4358 \, cm \][/tex]
- For the upper bound side length [tex]\( 15.5720 \, cm \)[/tex]:
[tex]\[ P_{\text{upper}} = 3 \times 15.5720 \][/tex]
[tex]\[ P_{\text{upper}} \approx 46.7160 \, cm \][/tex]
### Conclusion
The lower and upper bounds of the perimeter of the equilateral triangle with the given area are approximately:
- Lower bound perimeter: [tex]\( 44.4358 \, cm \)[/tex]
- Upper bound perimeter: [tex]\( 46.7160 \, cm \)[/tex]
Thus, the lower bound of the perimeter is approximately [tex]\( 44.436 \, cm \)[/tex], and the upper bound of the perimeter is approximately [tex]\( 46.716 \, cm \)[/tex].
### Step 1: Determine the range for the area
Given that the area of the triangle is measured to the nearest [tex]\( 10 \, cm^2 \)[/tex], the actual area can vary within a range of [tex]\( \pm 5 \, cm^2 \)[/tex] from 100 [tex]\( cm^2 \)[/tex]. Therefore:
- The lower bound for the area is [tex]\( 100 - 5 = 95 \, cm^2 \)[/tex]
- The upper bound for the area is [tex]\( 100 + 5 = 105 \, cm^2 \)[/tex]
### Step 2: Recall the formula for the area of an equilateral triangle
The area [tex]\( A \)[/tex] of an equilateral triangle with side length [tex]\( s \)[/tex] is given by:
[tex]\[ A = \frac{\sqrt{3}}{4} s^2 \][/tex]
### Step 3: Find the side lengths for the lower and upper bounds of the area
- For the lower bound area of [tex]\( 95 \, cm^2 \)[/tex]:
[tex]\[ 95 = \frac{\sqrt{3}}{4} s_{\text{lower}}^2 \][/tex]
[tex]\[ s_{\text{lower}}^2 = \frac{95 \times 4}{\sqrt{3}} \][/tex]
[tex]\[ s_{\text{lower}} = \sqrt{\frac{380}{\sqrt{3}}} \][/tex]
[tex]\[ s_{\text{lower}} \approx 14.8119 \, cm \][/tex]
- For the upper bound area of [tex]\( 105 \, cm^2 \)[/tex]:
[tex]\[ 105 = \frac{\sqrt{3}}{4} s_{\text{upper}}^2 \][/tex]
[tex]\[ s_{\text{upper}}^2 = \frac{105 \times 4}{\sqrt{3}} \][/tex]
[tex]\[ s_{\text{upper}} = \sqrt{\frac{420}{\sqrt{3}}} \][/tex]
[tex]\[ s_{\text{upper}} \approx 15.5720 \, cm \][/tex]
### Step 4: Calculate the perimeter for the lower and upper bounds
The perimeter [tex]\( P \)[/tex] of an equilateral triangle with side length [tex]\( s \)[/tex] is given by:
[tex]\[ P = 3s \][/tex]
- For the lower bound side length [tex]\( 14.8119 \, cm \)[/tex]:
[tex]\[ P_{\text{lower}} = 3 \times 14.8119 \][/tex]
[tex]\[ P_{\text{lower}} \approx 44.4358 \, cm \][/tex]
- For the upper bound side length [tex]\( 15.5720 \, cm \)[/tex]:
[tex]\[ P_{\text{upper}} = 3 \times 15.5720 \][/tex]
[tex]\[ P_{\text{upper}} \approx 46.7160 \, cm \][/tex]
### Conclusion
The lower and upper bounds of the perimeter of the equilateral triangle with the given area are approximately:
- Lower bound perimeter: [tex]\( 44.4358 \, cm \)[/tex]
- Upper bound perimeter: [tex]\( 46.7160 \, cm \)[/tex]
Thus, the lower bound of the perimeter is approximately [tex]\( 44.436 \, cm \)[/tex], and the upper bound of the perimeter is approximately [tex]\( 46.716 \, cm \)[/tex].
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