A swimmer heads directly across a river, swimming at 1.72 m/s relative to the water. She arrives at a point 39.0 m downstream from the point directly across the river, which is 72.7 m wide. What is the speed of the river current?

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To find the speed of the river current, we can use the following steps:

Step 1: Identify the given information.
- The swimmer's speed relative to the water (v_swimmer) is 1.72 m/s.
- The distance the swimmer drifts downstream (d_downstream) is 39.0 m.
- The width of the river (d_river) is 72.7 m.

Step 2: Identify the unknown variable.
- The unknown variable is the speed of the river current (v_current).

Step 3: Set up the equation.
- We can use the equation: v_swimmer = √(v_current^2 + v_perpendicular^2), where v_perpendicular is the component of the swimmer's velocity perpendicular to the current.

Step 4: Calculate the perpendicular component of velocity.
- The perpendicular component of velocity is the distance the swimmer drifts downstream divided by the time it takes to cross the river.
- The time it takes to cross the river (t_cross) is given by the width of the river divided by the swimmer's velocity relative to the water: t_cross = d_river / v_swimmer.
- Therefore, v_perpendicular = d_downstream / t_cross = d_downstream / (d_river / v_swimmer) = (d_downstream * v_swimmer) / d_river.

Step 5: Substitute the values into the equation.
- v_swimmer = √(v_current^2 + ((d_downstream * v_swimmer) / d_river)^2).

Step 6: Rearrange the equation to solve for v_current.
- Subtract v_current^2 from both sides: v_swimmer^2 - v_current^2 = (d_downstream * v_swimmer / d_river)^2.
- Take the square root of both sides: √(v_swimmer^2 - v_current^2) = d_downstream * v_swimmer / d_river.
- Multiply both sides by d_river: d_river * √(v_swimmer^2 - v_current^2) = d_downstream * v_swimmer.
- Divide both sides by v_swimmer: d_river * √(v_swimmer^2 - v_current^2) / v_swimmer = d_downstream.
- Multiply both sides by v_swimmer: d_river * √(v_swimmer^2 - v_current^2) = d_downstream * v_swimmer.
- Divide both sides by d_river: √(v_swimmer^2 - v_current^2) = (d_downstream * v_swimmer) / d_river.

Step 7: Solve for v_current.
- Square both sides of the equation: v_swimmer^2 - v_current^2 = (d_downstream * v_swimmer)^2 / d_river^2.
- Rearrange the equation: v_swimmer^2 = v_current^2 + (d_downstream * v_swimmer)^2 / d_river^2.
- Subtract (d_downstream * v_swimmer)^2 / d_river^2 from both sides: v_swimmer^2 - (d_downstream * v_swimmer)^2 / d_river^2 = v_current^2.
- Take the square root of both sides: v_current = √(v_swimmer^2 - (d_downstream * v_swimmer)^2 / d_river^2).

Step 8: Substitute the given values into the equation and calculate.
- v_current = √(1.72^2 - (39.0 * 1.72)^2 / 72.7^2).

Using a calculator, we find that the speed of the river current is approximately 0.96 m/s.

To find the speed of the river current, we need to analyze the swimmer's motion relative to the shore. Here's how we can do it:

1. Let's assume the swimmer's velocity relative to the shore is v_swimmer.
2. The swimmer's velocity relative to the water is given as 1.72 m/s.
3. The river current's velocity is given as v_current.
4. Since the swimmer is heading directly across the river, the component of the swimmer's velocity perpendicular to the river's current will cancel each other out. Therefore, v_swimmer will be equal to (-v_current) + 1.72 m/s.
5. We can write an equation to relate the distances traveled by the swimmer relative to the shore and the water:
distance traveled relative to the shore = distance traveled relative to the water + distance due to the river's current.
39.0 m = 1.72 m/s × t + 72.7 m × t (where t is the time taken to cross the river).
6. Solving the equation for t, we get t = 39.0 m / (1.72 m/s + 72.7 m).
7. Substitute the values into the equation and solve for t.
8. Once we know the value of t, we can substitute it back into the equation v_swimmer = (-v_current) + 1.72 m/s and solve for v_current.

Let's calculate it step by step:

39.0 m = (1.72 m/s + v_current) * t + 72.7 m * t

Rearranging the equation, we get:
39.0 m - 72.7 m * t = (1.72 m/s + v_current) * t

Now solve for t:
39.0 m - 72.7 m * t = 1.72 m/s * t + v_current * t

39.0 m = (1.72 m/s + v_current) * t + 72.7 m * t

39.0 m - 72.7 m * t = 1.72 m/s * t + v_current * t

Now, rearrange the equation and solve for t:

39.0 m - 72.7 m * t = 1.72 m/s * t + v_current * t

Subtract v_current * t from both sides:

39.0 m - 72.7 m * t - v_current * t = 1.72 m/s * t

Combine like terms:

39.0 m = (1.72 m/s + 72.7 m + v_current) * t

Divide both sides by (1.72 m/s + 72.7 m + v_current) :

t = 39.0 m / (1.72 m/s + 72.7 m + v_current)

Now that we have the value of t, let's substitute it back into the equation v_swimmer = (-v_current) + 1.72 m/s:

v_swimmer = (-v_current) + 1.72 m/s

Finally, substitute the value of v_swimmer as 1.72 m/s and solve for v_current:

1.72 m/s = (-v_current) + 1.72 m/s

Simplifying the equation, we find:

v_current = 0 m/s

Therefore, the speed of the river current is 0 m/s, meaning there is no current.