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Sagot :
To help Helen find the value of [tex]\(\pi\)[/tex] and its bounds, we'll start with the given measurements: a circumference [tex]\(C = 405\)[/tex] mm (correct to 3 significant figures) and a diameter [tex]\(d = 130\)[/tex] mm (correct to 2 significant figures). She wants to calculate [tex]\(\pi\)[/tex] using the formula:
[tex]\[ \pi = \frac{C}{d} \][/tex]
First, we find the estimated value of [tex]\(\pi\)[/tex]:
[tex]\[ \pi = \frac{405}{130} \approx 3.115 \][/tex]
Next, we need to determine the lower and upper bounds for the circumference and the diameter, then use these bounds to calculate the lower and upper bounds for [tex]\(\pi\)[/tex].
### Lower and Upper Bounds of Measurements
Since these measurements are given to 3 and 2 significant figures respectively, we can deduce:
- For the circumference [tex]\(C = 405\)[/tex] mm:
- Lower bound: [tex]\(404.5\)[/tex] mm
- Upper bound: [tex]\(405.5\)[/tex] mm
- For the diameter [tex]\(d = 130\)[/tex] mm:
- Lower bound: [tex]\(125\)[/tex] mm (slightly rounded down for a safe lower bound at 2 significant figures)
- Upper bound: [tex]\(135\)[/tex] mm (slightly rounded up for a safe upper bound at 2 significant figures)
### Calculating Bounds for [tex]\(\pi\)[/tex]
#### Lower Bound for [tex]\(\pi\)[/tex]:
To get the lower bound for [tex]\(\pi\)[/tex], divide the lower bound of circumference by the upper bound of diameter:
[tex]\[ \pi_{\text{lower}} = \frac{404.5}{135} \approx 2.996 \][/tex]
#### Upper Bound for [tex]\(\pi\)[/tex]:
To get the upper bound for [tex]\(\pi\)[/tex], divide the upper bound of circumference by the lower bound of diameter:
[tex]\[ \pi_{\text{upper}} = \frac{405.5}{125} \approx 3.244 \][/tex]
### Final Results
Rounds the values to 3 decimal places:
- Estimated value of [tex]\(\pi\)[/tex]: [tex]\(3.115\)[/tex]
- Lower bound for [tex]\(\pi\)[/tex]: [tex]\(2.996\)[/tex]
- Upper bound for [tex]\(\pi\)[/tex]: [tex]\(3.244\)[/tex]
Thus, Helen's calculated value of [tex]\(\pi\)[/tex] is approximately [tex]\(3.115\)[/tex] with bounds:
[tex]\[ \pi_{\text{lower}} = 2.996 \quad \text{and} \quad \pi_{\text{upper}} = 3.244 \][/tex]
So, the value of [tex]\(\pi\)[/tex] Helen gets using her measurements is [tex]\(3.115\)[/tex] bounded by [tex]\(2.996\)[/tex] and [tex]\(3.244\)[/tex], all to 3 decimal places.
[tex]\[ \pi = \frac{C}{d} \][/tex]
First, we find the estimated value of [tex]\(\pi\)[/tex]:
[tex]\[ \pi = \frac{405}{130} \approx 3.115 \][/tex]
Next, we need to determine the lower and upper bounds for the circumference and the diameter, then use these bounds to calculate the lower and upper bounds for [tex]\(\pi\)[/tex].
### Lower and Upper Bounds of Measurements
Since these measurements are given to 3 and 2 significant figures respectively, we can deduce:
- For the circumference [tex]\(C = 405\)[/tex] mm:
- Lower bound: [tex]\(404.5\)[/tex] mm
- Upper bound: [tex]\(405.5\)[/tex] mm
- For the diameter [tex]\(d = 130\)[/tex] mm:
- Lower bound: [tex]\(125\)[/tex] mm (slightly rounded down for a safe lower bound at 2 significant figures)
- Upper bound: [tex]\(135\)[/tex] mm (slightly rounded up for a safe upper bound at 2 significant figures)
### Calculating Bounds for [tex]\(\pi\)[/tex]
#### Lower Bound for [tex]\(\pi\)[/tex]:
To get the lower bound for [tex]\(\pi\)[/tex], divide the lower bound of circumference by the upper bound of diameter:
[tex]\[ \pi_{\text{lower}} = \frac{404.5}{135} \approx 2.996 \][/tex]
#### Upper Bound for [tex]\(\pi\)[/tex]:
To get the upper bound for [tex]\(\pi\)[/tex], divide the upper bound of circumference by the lower bound of diameter:
[tex]\[ \pi_{\text{upper}} = \frac{405.5}{125} \approx 3.244 \][/tex]
### Final Results
Rounds the values to 3 decimal places:
- Estimated value of [tex]\(\pi\)[/tex]: [tex]\(3.115\)[/tex]
- Lower bound for [tex]\(\pi\)[/tex]: [tex]\(2.996\)[/tex]
- Upper bound for [tex]\(\pi\)[/tex]: [tex]\(3.244\)[/tex]
Thus, Helen's calculated value of [tex]\(\pi\)[/tex] is approximately [tex]\(3.115\)[/tex] with bounds:
[tex]\[ \pi_{\text{lower}} = 2.996 \quad \text{and} \quad \pi_{\text{upper}} = 3.244 \][/tex]
So, the value of [tex]\(\pi\)[/tex] Helen gets using her measurements is [tex]\(3.115\)[/tex] bounded by [tex]\(2.996\)[/tex] and [tex]\(3.244\)[/tex], all to 3 decimal places.
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