Edits to “Statement”, “Solution”, “Answer”

astrosander edited Dec 9, 2025, 2:04 PM

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revision #15875 parent #15855 ← older

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		+	### Statement
		+
		+	$1.1.10^*.$ 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 $u = v/2$.
		+
		+	![ For problem $1.1.10^*$ \|604x409, 39%](../../img/1.1.10/statement.png)
		+
		+	### Solution
		+
		+	![ Chasing a bus \|632x365, 37%](../../img/1.1.10/1.1.10.png)
		+
		+	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: $AC = L, BC = h, AB = s.$
		+
		+	![ The plane bounded by $\alpha$ \|349x306, 34%](../../img/1.1.10/sol.png)
		+
		+	2\. From right-angled triangle $ABC$ we have
		+
		+	$$
		+	L = s \cdot \cos\frac{\alpha}{2};\quad h = s \cdot\sin\frac{\alpha}{2}
		+	$$
		+
		+	3\. Travel time of bus $t_1$ and passenger $t_2$ before meeting at point $C$
		+
		+	$$
		+	{t}_{1}=\frac{{L}}{{v}}=\frac{{s}\cos(\alpha /2)}{{v}};\quad{t}_{2}=\frac{{h}}{{u}}=\frac{{s}\sin(\alpha /2)}{{u}}
		+	$$
		+
		+	where from
		+
		+	$$
		+	\fbox{$\alpha = 2 \cdot \arctan \frac{u}{v}$}
		+	$$
		+
		+	#### Answer
		+
		+	From the region bounded by the angle $\alpha = 2 \arctan (u/v)$ with vertex at the point $A$, bisected by the motorway

	@@ -1,1 +1,35 @@
	@@sUf04		### Statement

			$1.1.10^*.$ 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 $u = v/2$.

			![ For problem $1.1.10^*$ \|604x409, 39%](../../img/1.1.10/statement.png)

			### Solution

			![ Chasing a bus \|632x365, 37%](../../img/1.1.10/1.1.10.png)

			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: $AC = L, BC = h, AB = s.$

			![ The plane bounded by $\alpha$ \|349x306, 34%](../../img/1.1.10/sol.png)

			2\. From right-angled triangle $ABC$ we have

			$$
			L = s \cdot \cos\frac{\alpha}{2};\quad h = s \cdot\sin\frac{\alpha}{2}
			$$

			3\. Travel time of bus $t_1$ and passenger $t_2$ before meeting at point $C$

			$$
			{t}_{1}=\frac{{L}}{{v}}=\frac{{s}\cos(\alpha /2)}{{v}};\quad{t}_{2}=\frac{{h}}{{u}}=\frac{{s}\sin(\alpha /2)}{{u}}
			$$

			where from

			$$
			\fbox{$\alpha = 2 \cdot \arctan \frac{u}{v}$}
			$$

			#### Answer

			From the region bounded by the angle $\alpha = 2 \arctan (u/v)$ with vertex at the point $A$, bisected by the motorway