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Sagot :
To find the lower bound and upper bound for Helen's value of [tex]\(\pi\)[/tex], we need to consider the precision of the given measurements for the circumference ([tex]\(C\)[/tex]) and the diameter ([tex]\(d\)[/tex]).
The measured values are:
- Circumference [tex]\(C = 405 \, \text{mm}\)[/tex], correct to 3 significant figures.
- Diameter [tex]\(d = 130 \, \text{mm}\)[/tex], correct to 2 significant figures.
### Step 1: Calculate Helen's measured value of [tex]\(\pi\)[/tex]
First, Helen's formula for [tex]\(\pi\)[/tex] is:
[tex]\[ \pi = \frac{C}{d} \][/tex]
Plugging in the measured values:
[tex]\[ \pi = \frac{405}{130} \approx 3.115 \][/tex]
### Step 2: Determine the bounds for circumference and diameter
Since [tex]\(C = 405 \, \text{mm}\)[/tex] is correct to 3 significant figures, the bounds for [tex]\(C\)[/tex] are:
[tex]\[ \text{Lower bound of } C = 404.5 \, \text{mm} \][/tex]
[tex]\[ \text{Upper bound of } C = 405.5 \, \text{mm} \][/tex]
Since [tex]\(d = 130 \, \text{mm}\)[/tex] is correct to 2 significant figures, the bounds for [tex]\(d\)[/tex] are:
[tex]\[ \text{Lower bound of } d = 125 \, \text{mm} \][/tex]
[tex]\[ \text{Upper bound of } d = 135 \, \text{mm} \][/tex]
### Step 3: Calculate the lower bound for [tex]\(\pi\)[/tex]
To find the lower bound for [tex]\(\pi\)[/tex], use the smallest possible circumference and the largest possible diameter:
[tex]\[ \pi_{\text{lower bound}} = \frac{\text{Lower bound of } C}{\text{Upper bound of } d} = \frac{404.5}{135} \][/tex]
### Step 4: Calculate the upper bound for [tex]\(\pi\)[/tex]
To find the upper bound for [tex]\(\pi\)[/tex], use the largest possible circumference and the smallest possible diameter:
[tex]\[ \pi_{\text{upper bound}} = \frac{\text{Upper bound of } C}{\text{Lower bound of } d} = \frac{405.5}{125} \][/tex]
### Step 5: Determine the values
Carrying out these divisions gives:
[tex]\[ \pi_{\text{lower bound}} = \frac{404.5}{135} \approx 2.996 \][/tex]
[tex]\[ \pi_{\text{upper bound}} = \frac{405.5}{125} \approx 3.244 \][/tex]
### Conclusion
The lower bound and upper bound for Helen's value of [tex]\(\pi\)[/tex] are:
[tex]\[ \pi_{\text{lower bound}} \approx 2.996 \][/tex]
[tex]\[ \pi_{\text{upper bound}} \approx 3.244 \][/tex]
Therefore, the lower bound for [tex]\(\pi\)[/tex] is [tex]\(2.996\)[/tex] and the upper bound for [tex]\(\pi\)[/tex] is [tex]\(3.244\)[/tex], both correct to 3 decimal places.
The measured values are:
- Circumference [tex]\(C = 405 \, \text{mm}\)[/tex], correct to 3 significant figures.
- Diameter [tex]\(d = 130 \, \text{mm}\)[/tex], correct to 2 significant figures.
### Step 1: Calculate Helen's measured value of [tex]\(\pi\)[/tex]
First, Helen's formula for [tex]\(\pi\)[/tex] is:
[tex]\[ \pi = \frac{C}{d} \][/tex]
Plugging in the measured values:
[tex]\[ \pi = \frac{405}{130} \approx 3.115 \][/tex]
### Step 2: Determine the bounds for circumference and diameter
Since [tex]\(C = 405 \, \text{mm}\)[/tex] is correct to 3 significant figures, the bounds for [tex]\(C\)[/tex] are:
[tex]\[ \text{Lower bound of } C = 404.5 \, \text{mm} \][/tex]
[tex]\[ \text{Upper bound of } C = 405.5 \, \text{mm} \][/tex]
Since [tex]\(d = 130 \, \text{mm}\)[/tex] is correct to 2 significant figures, the bounds for [tex]\(d\)[/tex] are:
[tex]\[ \text{Lower bound of } d = 125 \, \text{mm} \][/tex]
[tex]\[ \text{Upper bound of } d = 135 \, \text{mm} \][/tex]
### Step 3: Calculate the lower bound for [tex]\(\pi\)[/tex]
To find the lower bound for [tex]\(\pi\)[/tex], use the smallest possible circumference and the largest possible diameter:
[tex]\[ \pi_{\text{lower bound}} = \frac{\text{Lower bound of } C}{\text{Upper bound of } d} = \frac{404.5}{135} \][/tex]
### Step 4: Calculate the upper bound for [tex]\(\pi\)[/tex]
To find the upper bound for [tex]\(\pi\)[/tex], use the largest possible circumference and the smallest possible diameter:
[tex]\[ \pi_{\text{upper bound}} = \frac{\text{Upper bound of } C}{\text{Lower bound of } d} = \frac{405.5}{125} \][/tex]
### Step 5: Determine the values
Carrying out these divisions gives:
[tex]\[ \pi_{\text{lower bound}} = \frac{404.5}{135} \approx 2.996 \][/tex]
[tex]\[ \pi_{\text{upper bound}} = \frac{405.5}{125} \approx 3.244 \][/tex]
### Conclusion
The lower bound and upper bound for Helen's value of [tex]\(\pi\)[/tex] are:
[tex]\[ \pi_{\text{lower bound}} \approx 2.996 \][/tex]
[tex]\[ \pi_{\text{upper bound}} \approx 3.244 \][/tex]
Therefore, the lower bound for [tex]\(\pi\)[/tex] is [tex]\(2.996\)[/tex] and the upper bound for [tex]\(\pi\)[/tex] is [tex]\(3.244\)[/tex], both correct to 3 decimal places.
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