What Are the Solutions to These River Current Boat Problems?

[NoPhysicsGenius]

[SOLVED] Boat in the river problems

Homework Statement

(a.) A boat took 1 hour 50 minutes to go 55 miles downstream and 3 hours 40 minutes to return. Find the rate of the current.

(b.) A boat travels 60 miles downstream in the same time it takes to go 36 miles upstream. The speed of the boat is 15 mi/h greater than the speed of the current. Find the speed of the current.

(c.) A boat travels 12 miles downstream in 1.5 hours. On the return trip the boat travels the same distance upstream in 2 hours. Find the rate of the boat in still water and the rate of the current.

Homework Equations

distance = rate * time

The Attempt at a Solution

I can't figure out how to solve any of these. Could you at least help me to set them up correctly? Thank you.

 

[Nate810]

(a.) Let x be the rate of the current. We have 55/x + (55/x -15) = 5.5 + 6.667 55/x = 11.167 x = 5 mph (b.) Let y be the speed of the current. We have 60/(y+15) + 36/(y-15) = 4 + 3 96/y = 7 y = 13.7 mph (c.) Let z be the rate of the boat in still water and a be the rate of the current.We have 12/z + 12/(z+a) = 1.5 + 2 24/z = 3.5 z = 6.86 mph and 12/(z-a) + 12/z = 2 + 1.5 24/z = 3.5 a = 3.14 mph

 

[Moetasim]

Sure, let's work through these problems together. Let's start with problem (a).

(a) In this problem, we are given the distance (55 miles) and the time it took for the boat to travel downstream (1 hour 50 minutes = 1.83 hours) and upstream (3 hours 40 minutes = 3.67 hours). We are looking for the rate of the current.

To solve this problem, we can use the formula distance = rate * time. For the downstream trip, we can write the equation as 55 = (rate of boat + rate of current) * 1.83. Similarly, for the upstream trip, we can write the equation as 55 = (rate of boat - rate of current) * 3.67.

Now, we have two equations with two unknowns (rate of boat and rate of current). We can solve for either variable by using substitution or elimination. Let's use substitution.

From the first equation, we can rearrange it to get the rate of boat + rate of current = 30.05. We can then substitute this value into the second equation, giving us 55 = (30.05 - rate of current) * 3.67. Simplifying this equation, we get 55 = 110.35 - 3.67(rate of current). Solving for the rate of current, we get a value of approximately 8.58 mph.

(b) In this problem, we are given the distance (60 miles downstream and 36 miles upstream) and the relationship between the boat's speed and the current's speed (boat's speed = current's speed + 15 mph). We are looking for the speed of the current.

Again, we can use the formula distance = rate * time to set up our equations. For the downstream trip, we can write the equation as 60 = (rate of boat + rate of current) * t, where t is the time it took for the boat to travel downstream. Similarly, for the upstream trip, we can write the equation as 36 = (rate of boat - rate of current) * t.

Since we are given that the boat's speed is 15 mph greater than the current's speed, we can rewrite the equations as 60 = (rate of current + 15 + rate of current) * t and 36 = (rate of current + 15 - rate of current) *