Certainly! Let's break down the solution step-by-step in order to find the lower and upper bounds for the expression [tex]\(\frac{r}{t-v}\)[/tex].
### Step 1: Determine the bounds for [tex]\(r\)[/tex]
Given [tex]\(r = 380\)[/tex] rounded to 2 significant figures:
- The lower bound of [tex]\(r\)[/tex] is [tex]\(375\)[/tex].
- The upper bound of [tex]\(r\)[/tex] is [tex]\(385\)[/tex].
### Step 2: Determine the bounds for [tex]\(t\)[/tex]
Given [tex]\(t = 24\)[/tex] rounded to the nearest integer:
- The lower bound of [tex]\(t\)[/tex] is [tex]\(23.5\)[/tex].
- The upper bound of [tex]\(t\)[/tex] is [tex]\(24.5\)[/tex].
### Step 3: Determine the bounds for [tex]\(v\)[/tex]
Given [tex]\(v = 4.6\)[/tex] rounded to 1 decimal place:
- The lower bound of [tex]\(v\)[/tex] is [tex]\(4.55\)[/tex].
- The upper bound of [tex]\(v\)[/tex] is [tex]\(4.65\)[/tex].
### Step 4: Calculate the bounds for the divisor [tex]\(t - v\)[/tex]
To find the bounds for [tex]\(t - v\)[/tex], we need to consider the possible range:
- The lower bound of [tex]\(t - v\)[/tex] is calculated using the lower bound of [tex]\(t\)[/tex] and the upper bound of [tex]\(v\)[/tex]:
[tex]\[
t_{\text{lower}} - v_{\text{upper}} = 23.5 - 4.65 = 18.85
\][/tex]
- The upper bound of [tex]\(t - v\)[/tex] is calculated using the upper bound of [tex]\(t\)[/tex] and the lower bound of [tex]\(v\)[/tex]:
[tex]\[
t_{\text{upper}} - v_{\text{lower}} = 24.5 - 4.55 = 19.95
\][/tex]
### Step 5: Calculate the bounds for [tex]\(\frac{r}{t-v}\)[/tex]
For the expression [tex]\(\frac{r}{t-v}\)[/tex]:
- The lower bound of [tex]\(\frac{r}{t-v}\)[/tex] is calculated using the lower bound of [tex]\(r\)[/tex] and the upper bound of the divisor:
[tex]\[
\frac{r_{\text{lower}}}{(t-v)_{\text{upper}}} = \frac{375}{19.95} \approx 18.8
\][/tex]
- The upper bound of [tex]\(\frac{r}{t-v}\)[/tex] is calculated using the upper bound of [tex]\(r\)[/tex] and the lower bound of the divisor:
[tex]\[
\frac{r_{\text{upper}}}{(t-v)_{\text{lower}}} = \frac{385}{18.85} \approx 20.4
\][/tex]
### Step 6: Final Result
The lower bound of [tex]\(\frac{r}{t-v}\)[/tex] is [tex]\(18.8\)[/tex], and the upper bound of [tex]\(\frac{r}{t-v}\)[/tex] is [tex]\(20.4\)[/tex], both rounded to 1 decimal place.
Thus, the lower and upper bounds for [tex]\(\frac{r}{t-v}\)[/tex] are:
[tex]\[
\boxed{18.8 \text{ to } 20.4}
\][/tex]
