Two roads meeting a river at different angles (heights and distances)

In summary: The length of AC will always be the same, no matter where the two roads cross the river.It's been known to happen that the posted answers in a textbook are wrong ...Yes, forgive me. AC remains the same on either side. It is a puzzle. I think I have to accept the book being mistaken here.
  • #1
brotherbobby
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Homework Statement
Two straight roads, which cross one another, meet a river with straight course at angles ##60^{\circ}## and ##30^{\circ}##, respectively. If it be 3 miles by the longer of the two roads, from the crossing to the river, how far is it by the ##\text{shortest}## (or the shorter of the two?)? If there be a foot-path which goes the shortest way from the crossing to the river, what is the distance by it?
Relevant Equations
1. In a right-angled triangle, ##\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}##.
2. For a triangle ##\text{ABC}##, ##\frac{a}{\sin A} = \frac{b}{\sin B}##. (The Law of Sines)
1678645745230.pngProblem statement : I copy and paste the problem as it appears in the text. I hope am understanding its wording correctly.
1678647136482.png
Diagram : The river is shown in blue. The two roads start from the crossing C and end in A and B, making angles ##60^{\circ}## and ##30^{\circ}## resectively. The longer of the two roads CB is given to be 3 miles. Question is, how long is the shorter of the two roads AC, assuming my understanding of the problem is correct. Additionally, how long is the shortest way from C to the river, that is CD, where CD is ##\perp^r## to BA produced.

Solution : Using the law of sines (see Relevant Equations above), ##\text{AC} = \text{BC}\times \frac{\sin 30^{\circ}}{\sin 60^{\circ}} = 3\times \frac{1}{2}\times \frac{2}{\sqrt 3} = \sqrt 3 = \boxed{1.73\,\, \text{miles}}##.
The distance CD = ##BC\, \sin 30^{\circ} = 3\times\frac{1}{2} = \boxed{1.5\,\text{miles}}##.Issue : The answers don't match with those in the text. I copy and paste the text answers below.
1678648034873.png

A hint or help would be welcome. I am perplexed at the moment, given the simplicity of the problem. Or am I deceiving myself?
 
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  • #2
I get the same answer as you got. For the pasted textbook answer, I don't know how they got ##2 \frac 1 2## miles, and I don't understand the first part, which is unclear. It says either "##2 \cdot 89##... miles" or "##2 \cdot 80##... miles".
 
  • #3
Mark44 said:
I get the same answer as you got. For the pasted textbook answer, I don't know how they got ##2 \frac 1 2## miles, and I don't understand the first part, which is unclear. It says either "##2 \cdot 89##... miles" or "##2 \cdot 80##... miles".
I think I have an idea. I assumed (naively) that the two roads "meet" the river bank on the same side as that of the perpendicular (CD). Is it possible that they meet on opposite sides of D? I should carry out the solution if that is what the authors mean by "crossing".

All the same, the wording is ambiguous, if you'll agree.
 
  • #4
brotherbobby said:
Is it possible that they meet on opposite sides of D?
I thought of that, too, but I don't think it makes a difference in the length of AC.
 
  • #5
Yes, forgive me. AC remains the same on either side. It is a puzzle. I think I have to accept the book being mistaken here.
 
  • #6
brotherbobby said:
Yes, forgive me. AC remains the same on either side. It is a puzzle. I think I have to accept the book being mistaken here.
It's been known to happen that the posted answers in a textbook are wrong ...
 
  • #7
brotherbobby said:
Yes, forgive me. AC remains the same on either side. It is a puzzle. I think I have to accept the book being mistaken here.
I believe the mistake is in the 3 miles distance.
For the values of the response to be correct, the distance from crossing to river via BC should have been 5.0 miles.
 
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  • #8
You don't even need the sine rule to solve this problem since we are dealing with right triangles.
 
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FAQ: Two roads meeting a river at different angles (heights and distances)

What is the optimal way to construct a bridge over a river where two roads meet at different angles?

The optimal way to construct a bridge over a river where two roads meet at different angles depends on minimizing the total construction cost, which includes the cost of the bridge and the connecting roads. Engineers often use geometric and algebraic methods to determine the shortest and most cost-effective path for the bridge, taking into account factors such as the angles of the roads, the width of the river, and the topography of the surrounding area.

How do you calculate the length of a bridge needed when roads meet a river at different heights?

To calculate the length of a bridge needed when roads meet a river at different heights, you can use trigonometric principles. First, determine the horizontal distance between the two points where the roads meet the river. Then, use the height difference and the horizontal distance to calculate the hypotenuse of the right triangle formed, which represents the length of the bridge. The formula is: Length of Bridge = √(horizontal distance² + height difference²).

What factors should be considered when designing a bridge for roads meeting a river at different distances?

When designing a bridge for roads meeting a river at different distances, several factors should be considered, including the width of the river, the angles at which the roads approach the river, the height differences between the roads, soil and foundation conditions, environmental impact, cost, and the type of bridge that is most suitable for the location (e.g., beam, arch, suspension). Engineers must also consider the load-bearing capacity and durability of the bridge.

How do environmental factors influence the construction of a bridge over a river?

Environmental factors play a significant role in the construction of a bridge over a river. These include the river's flow rate, seasonal variations in water level, potential for flooding, local wildlife and ecosystems, and weather conditions. Engineers must design the bridge to withstand these environmental conditions while minimizing ecological disruption. This may involve choosing materials resistant to corrosion, designing for flood resilience, and implementing measures to protect local wildlife.

What are the common challenges faced during the construction of bridges over rivers with varying road angles and heights?

Common challenges during the construction of bridges over rivers with varying road angles and heights include ensuring structural stability, managing the complex geometry of the bridge, dealing with difficult terrain and foundation conditions, coordinating construction in a potentially hazardous environment, and minimizing the impact on the river's ecosystem. Additionally, logistical challenges such as transporting materials and equipment to the site and ensuring worker safety are also significant concerns. Advanced engineering techniques and careful planning are essential to address these challenges.

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