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| <h3 id="back-link"><a href="../#1.1">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../#1.1">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.1.10^*.$ A bus is driving along a straight highway at constant speed $v$. You have noticed the bus when it was at some point $A$. From what area near the highway can you catch up with this bus if your running speed is $u < v$? Draw this area for $u = v/2$. | | $1.1.10^*.$ A bus is driving along a straight highway at constant speed $v$. You have noticed the bus when it was at some point $A$. From what area near the highway can you catch up with this bus if your running speed is $u < v$? Draw this area for $u = v/2$. |
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| <figcaption> | | <figcaption> |
| For problem $1.1.10^*$ | | For problem $1.1.10^*$ |
| </figcaption> | | </figcaption> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
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| <center> | | <center> |
| <figure> | | <figure> |
| <img src="1.1.10.png" alt="1.1.10" | | <img src="1.1.10.png" alt="1.1.10" |
| loading="lazy" width="200" /> | | loading="lazy" width="200" /> |
| <figcaption> | | <figcaption> |
| Chasing a bus | | Chasing a bus |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
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| <p> | | <p> |
| 1. Let the bus is at point $A$, and the catching up person starts from point $B$ and runs perpendicular to the roadway $AC$. Let us introduce the notations: $AC = L, BC = h, AB = s.$ | | 1. Let the bus is at point $A$, and the catching up person starts from point $B$ and runs perpendicular to the roadway $AC$. Let us introduce the notations: $AC = L, BC = h, AB = s.$ |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="sol.png" alt="1.1.10" | | <img src="sol.png" alt="1.1.10" |
| loading="lazy" width="200" /> | | loading="lazy" width="200" /> |
| <figcaption> | | <figcaption> |
| The plane bounded by $\alpha$ | | The plane bounded by $\alpha$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
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| <p> | | <p> |
| 2. From right-angled triangle $ABC$ we have | | 2. From right-angled triangle $ABC$ we have |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$L = s \cdot cos \frac{\alpha}{2}\text{ и }h = s \cdot sin \frac{\alpha}{2}$$ | | $$L = s \cdot cos \frac{\alpha}{2}\text{ и }h = s \cdot sin \frac{\alpha}{2}$$ |
| </p> | | </p> |
| <p> | | <p> |
| 3. Travel time of bus $t_1$ and passenger $t_2$ before meeting at point $C$ | | 3. Travel time of bus $t_1$ and passenger $t_2$ before meeting at point $C$ |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| ${t}_{1}=\frac{{L}}{{v}}=\frac{{s}\cos(\alpha/2)}{{v}};\quad{t}_{2}=\frac{{h}}{{u}}=\frac{{s}\sin(\alpha/2)}{{u}}$ | | ${t}_{1}=\frac{{L}}{{v}}=\frac{{s}\cos(\alpha/2)}{{v}};\quad{t}_{2}=\frac{{h}}{{u}}=\frac{{s}\sin(\alpha/2)}{{u}}$ |
| </p> | | </p> |
| <p> | | <p> |
| where from | | where from |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$\fbox{$\alpha = 2 \cdot \text{arcsin} \frac{u}{v}$}$$ | | $$\fbox{$\alpha = 2 \cdot \text{arcsin} \frac{u}{v}$}$$ |
| </p> | | </p> |
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| </p> | | </p> |
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| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| From the region bounded by the angle $α = 2 \, \text{arcsin}(u/v)$ with vertex at the point $A$, bisected by the motorway | | From the region bounded by the angle $α = 2 \, \text{arcsin}(u/v)$ with vertex at the point $A$, bisected by the motorway |
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