A pole has to be erected at a point on the boundary of a circular park of diameter 13 metres in such a way that the difference of its distances from two diametrically opposite fixe... A pole has to be erected at a point on the boundary of a circular park of diameter 13 metres in such a way that the difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 metres. Is it possible to do so? If yes, at what distances from the two gates should the pole be erected? (or) A passenger train takes 3 hours less for a journey of 360 km if the speed is increased by 10 km/h from its usual speed. Find its usual speed.
Understand the Problem
The question asks if it is possible to erect a pole on the boundary of a circular park such that the difference in distances from two fixed gates A and B is 7 meters. If possible, it also asks for the specific distances from the gates where the pole should be erected. Additionally, it presents a different scenario involving a train's speed and time, asking for a calculation of its usual speed based on a change in travel conditions.
Answer
Yes, the pole can be erected. Distances: $6$ meters (A), $13$ meters (B) or $0$ meters (A), $7$ meters (B).
Answer for screen readers
Yes, it is possible to erect the pole. The specific distances are:
- From gate A: $6$ meters, from gate B: $13$ meters.
or
- From gate A: $0$ meters, from gate B: $7$ meters.
Steps to Solve
-
Identify the Geometry Involved We have a circular park with a diameter of 13 meters. The radius, $r$, is half of the diameter: $$ r = \frac{13}{2} = 6.5 \text{ meters} $$
-
Set Up the Distance Between Gates Let the two gates, A and B, be diametrically opposite each other on the boundary of the circle. The distance between them is equal to the diameter (d): $$ d = 13 \text{ meters} $$
-
Use the Condition of Distance Difference We want the difference in distances from the pole (point P) to the two gates (distance PA and distance PB) to be 7 meters: $$ |PA - PB| = 7 $$
-
Express the Distances from the Pole to Gates Let the distance from point P to gate A be $x$; therefore, the distance to gate B is: $$ PB = PA + 7 = x + 7 $$
-
Apply the Triangle Inequality Theorem Given that both distances must be less than or equal to the diameter of the circle:
-
( PA ) must be less than or equal to the diameter: $$ x \leq 13 $$
-
( PB ) must also be less than or equal to the diameter: $$ x + 7 \leq 13 $$
-
Solve the Inequalities From the second inequality, we can solve for (x): $$ x + 7 \leq 13 $$ $$ x \leq 6 $$
-
Analyze Possible Values for x So now we have:
- From distance to gate A: ( x \leq 13 )
- From distance to gate B: ( x \leq 6 )
The feasible range for (x) is: $$ 0 \leq x \leq 6 $$
- Find Specific Distances from Gates If we take a value (x = 6):
- For gate A, (PA = 6) meters
- For gate B, (PB = 6 + 7 = 13) meters
If we set (x = 0):
- For gate A, (PA = 0) meters
- For gate B, (PB = 0 + 7 = 7) meters
Thus, it is indeed possible to erect the pole, and the distances can be (6) meters from gate A and (13) meters from gate B, or (0) meters from gate A and (7) meters from gate B.
Yes, it is possible to erect the pole. The specific distances are:
- From gate A: $6$ meters, from gate B: $13$ meters.
or
- From gate A: $0$ meters, from gate B: $7$ meters.
More Information
In this problem, we applied geometric principles including the triangle inequality to find the feasible distances based on the given conditions. The difference in distances matches the specifications, demonstrating that the pole can be appropriately placed on the circular park boundary.
Tips
- Failing to apply the triangle inequality correctly, which can lead to ignoring the maximum distance limits.
- Misinterpreting the difference condition as a simple subtraction instead of considering both cases of addition and subtraction.
AI-generated content may contain errors. Please verify critical information
