Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Discover detailed solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
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].
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
