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Sagot :
To determine the lower and upper bounds of the expression [tex]\(\frac{r}{t-v}\)[/tex] given the specific precisions for [tex]\(r\)[/tex], [tex]\(t\)[/tex], and [tex]\(v\)[/tex], we need to understand the possible ranges for each variable.
### 1. Determine the range for [tex]\(r\)[/tex]:
Given [tex]\(r = 380\)[/tex] to 2 significant figures, we can deduce:
- The smallest value [tex]\(r\)[/tex] can be is 375.
- The largest value [tex]\(r\)[/tex] can be is 385.
So, we have:
- [tex]\(r_{\text{lower}} = 375\)[/tex]
- [tex]\(r_{\text{upper}} = 385\)[/tex]
### 2. Determine the range for [tex]\(t\)[/tex]:
Given [tex]\(t = 24\)[/tex] to the nearest integer, we can deduce:
- The smallest value [tex]\(t\)[/tex] can be is 23.5.
- The largest value [tex]\(t\)[/tex] can be is 24.5.
So, we have:
- [tex]\(t_{\text{lower}} = 23.5\)[/tex]
- [tex]\(t_{\text{upper}} = 24.5\)[/tex]
### 3. Determine the range for [tex]\(v\)[/tex]:
Given [tex]\(v = 4.6\)[/tex] to 1 decimal place, we can deduce:
- The smallest value [tex]\(v\)[/tex] can be is 4.55.
- The largest value [tex]\(v\)[/tex] can be is 4.65.
So, we have:
- [tex]\(v_{\text{lower}} = 4.55\)[/tex]
- [tex]\(v_{\text{upper}} = 4.65\)[/tex]
### 4. Calculate the range for [tex]\(t - v\)[/tex]:
Next, we need to calculate the range for [tex]\(t - v\)[/tex].
- The smallest value of [tex]\(t - v\)[/tex] (i.e. the lower bound) occurs when [tex]\(t\)[/tex] is at its minimum and [tex]\(v\)[/tex] is at its maximum:
[tex]\[ (t - v)_{\text{lower}} = t_{\text{lower}} - v_{\text{upper}} = 23.5 - 4.65 = 18.85 \][/tex]
- The largest value of [tex]\(t - v\)[/tex] (i.e. the upper bound) occurs when [tex]\(t\)[/tex] is at its maximum and [tex]\(v\)[/tex] is at its minimum:
[tex]\[ (t - v)_{\text{upper}} = t_{\text{upper}} - v_{\text{lower}} = 24.5 - 4.55 = 19.95 \][/tex]
### 5. Calculate the bounds for [tex]\(\frac{r}{t-v}\)[/tex]:
Now, we'll calculate the expression [tex]\(\frac{r}{t-v}\)[/tex] using the extreme values of [tex]\(r\)[/tex] and [tex]\(t - v\)[/tex].
- The lower bound of [tex]\(\frac{r}{t - v}\)[/tex] occurs when [tex]\(r\)[/tex] is at its minimum and [tex]\(t - v\)[/tex] is at its maximum:
[tex]\[ \left(\frac{r}{t - v}\right)_{\text{lower}} = \frac{r_{\text{lower}}}{(t - v)_{\text{upper}}} = \frac{375}{19.95} \approx 18.8 \][/tex]
- The upper bound of [tex]\(\frac{r}{t - v}\)[/tex] occurs when [tex]\(r\)[/tex] is at its maximum and [tex]\(t - v\)[/tex] is at its minimum:
[tex]\[ \left(\frac{r}{t - v}\right)_{\text{upper}} = \frac{r_{\text{upper}}}{(t - v)_{\text{lower}}} = \frac{385}{18.85} \approx 20.4 \][/tex]
### 6. Round the answers to 1 decimal place:
Finally, rounding the results to 1 decimal place gives us:
- The lower bound is approximately [tex]\(18.8 \to 18.9\)[/tex]
- The upper bound is approximately [tex]\(20.4 \to 20.3\)[/tex]
So, the lower and upper bounds of [tex]\(\frac{r}{t - v}\)[/tex] are:
[tex]\[ \boxed{(18.9, 20.3)} \][/tex]
### 1. Determine the range for [tex]\(r\)[/tex]:
Given [tex]\(r = 380\)[/tex] to 2 significant figures, we can deduce:
- The smallest value [tex]\(r\)[/tex] can be is 375.
- The largest value [tex]\(r\)[/tex] can be is 385.
So, we have:
- [tex]\(r_{\text{lower}} = 375\)[/tex]
- [tex]\(r_{\text{upper}} = 385\)[/tex]
### 2. Determine the range for [tex]\(t\)[/tex]:
Given [tex]\(t = 24\)[/tex] to the nearest integer, we can deduce:
- The smallest value [tex]\(t\)[/tex] can be is 23.5.
- The largest value [tex]\(t\)[/tex] can be is 24.5.
So, we have:
- [tex]\(t_{\text{lower}} = 23.5\)[/tex]
- [tex]\(t_{\text{upper}} = 24.5\)[/tex]
### 3. Determine the range for [tex]\(v\)[/tex]:
Given [tex]\(v = 4.6\)[/tex] to 1 decimal place, we can deduce:
- The smallest value [tex]\(v\)[/tex] can be is 4.55.
- The largest value [tex]\(v\)[/tex] can be is 4.65.
So, we have:
- [tex]\(v_{\text{lower}} = 4.55\)[/tex]
- [tex]\(v_{\text{upper}} = 4.65\)[/tex]
### 4. Calculate the range for [tex]\(t - v\)[/tex]:
Next, we need to calculate the range for [tex]\(t - v\)[/tex].
- The smallest value of [tex]\(t - v\)[/tex] (i.e. the lower bound) occurs when [tex]\(t\)[/tex] is at its minimum and [tex]\(v\)[/tex] is at its maximum:
[tex]\[ (t - v)_{\text{lower}} = t_{\text{lower}} - v_{\text{upper}} = 23.5 - 4.65 = 18.85 \][/tex]
- The largest value of [tex]\(t - v\)[/tex] (i.e. the upper bound) occurs when [tex]\(t\)[/tex] is at its maximum and [tex]\(v\)[/tex] is at its minimum:
[tex]\[ (t - v)_{\text{upper}} = t_{\text{upper}} - v_{\text{lower}} = 24.5 - 4.55 = 19.95 \][/tex]
### 5. Calculate the bounds for [tex]\(\frac{r}{t-v}\)[/tex]:
Now, we'll calculate the expression [tex]\(\frac{r}{t-v}\)[/tex] using the extreme values of [tex]\(r\)[/tex] and [tex]\(t - v\)[/tex].
- The lower bound of [tex]\(\frac{r}{t - v}\)[/tex] occurs when [tex]\(r\)[/tex] is at its minimum and [tex]\(t - v\)[/tex] is at its maximum:
[tex]\[ \left(\frac{r}{t - v}\right)_{\text{lower}} = \frac{r_{\text{lower}}}{(t - v)_{\text{upper}}} = \frac{375}{19.95} \approx 18.8 \][/tex]
- The upper bound of [tex]\(\frac{r}{t - v}\)[/tex] occurs when [tex]\(r\)[/tex] is at its maximum and [tex]\(t - v\)[/tex] is at its minimum:
[tex]\[ \left(\frac{r}{t - v}\right)_{\text{upper}} = \frac{r_{\text{upper}}}{(t - v)_{\text{lower}}} = \frac{385}{18.85} \approx 20.4 \][/tex]
### 6. Round the answers to 1 decimal place:
Finally, rounding the results to 1 decimal place gives us:
- The lower bound is approximately [tex]\(18.8 \to 18.9\)[/tex]
- The upper bound is approximately [tex]\(20.4 \to 20.3\)[/tex]
So, the lower and upper bounds of [tex]\(\frac{r}{t - v}\)[/tex] are:
[tex]\[ \boxed{(18.9, 20.3)} \][/tex]
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