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Given the values: [tex]\[ r = 380 \text{ (to 2 significant figures)} \][/tex] [tex]\[ t = 24 \text{ (to the nearest integer)} \][/tex] [tex]\[ v = 4.6 \text{ (to 1 decimal place)} \][/tex]
Work out the lower and upper bounds of [tex]\[ \frac{r}{t - v} \][/tex]
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:
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