Problem 91 - Precalculus with Calculus Previews

Question

Actual Direction The current in a river that is 0.5 mi across is \(6 \mathrm{mi} / \mathrm{h}\). A swimmer heads out from shore perpendicular to the current at \(2 \mathrm{mi} / \mathrm{h}\). In what direction is the swimmer actually going?

Step-by-Step Solution

Answer

The swimmer moves downstream at an angle \( \theta = \tan^{-1}(3) \) from perpendicular to the shore.

Step 1: Understand the Problem

We need to find the actual direction in which the swimmer is heading. The swimmer aims to swim perpendicular to the river flow, while the river's current pushes him sideways.

Step 2: Visualize the Vectors

Visualize the problem as a right-angled triangle where the swimmer's perpendicular velocity is one side and the river's current velocity is the other.

Step 3: Identify the Speeds

The swimmer's speed perpendicular to the river is \(2 \text{ mi/h}\). The river's current speed across the river is \(6 \text{ mi/h}\).

Step 4: Calculate the Resultant Velocity

The resultant velocity can be found using the Pythagorean theorem as it forms a right triangle.\[v = \sqrt{(6)^2 + (2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10}\]

Step 5: Calculate the Angle

The angle \( \theta \) with respect to the swimmer's initial heading can be calculated using the tangent function:\[\tan(\theta) = \frac{\text{river speed}}{\text{swimmer speed}} = \frac{6}{2} = 3\]\[\theta = \tan^{-1}(3)\]

Step 6: Determine the Actual Direction

The actual direction of the swimmer can be described as the angle \( \theta \) calculated from the perpendicular path the swimmer intended to take. Therefore, the swimmer goes downstream at an angle \( \theta \) from the line perpendicular to the shore.

Key Concepts

Resultant Velocity

Resultant velocity refers to the combination of two or more individual velocities. In this exercise, we are dealing with the swimmer's velocity and the river's current. When these two velocities combine, they create a resultant vector, which is the actual trajectory the swimmer follows.

Visualize this like an arrow diagram. One arrow represents the speed at which the swimmer heads across the river, while a second arrow indicates the force of the river pushing sideways. The resultant velocity is the diagonal arrow that forms between these two, pointing in the true direction the swimmer moves.

• Swimmer's velocity: 2 mi/h (perpendicular to river)
• River current velocity: 6 mi/h

To calculate the resultant velocity's magnitude, you can use the Pythagorean theorem, which we'll touch on next.

Pythagorean Theorem

The Pythagorean theorem helps us in calculating the magnitude of the resultant velocity in motion problems like this one. It states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

This theorem is particularly useful here because it lets us form a right triangle using our two vectors:

• Swimmer's velocity (one side of the triangle)
• River's current (the other side)

The diagonal (resultant velocity) is the hypotenuse. So, you can find the length of the hypotenuse with the equation:
\[ v = \sqrt{(6)^2 + (2)^2} = \sqrt{40} = 2\sqrt{10} \]
This tells you how fast the swimmer moves in the actual direction, influenced by both his own swimming power and the river's current.

Trigonometry in Motion Problems

Trigonometry is used in motion problems to find angles and directions, allowing us to understand how different forces combine. In this exercise, we want to discover the angle between the swimmer's intended path and his actual path.

You can determine this angle using trigonometric functions, specifically the tangent. When two sides of a right triangle are known, tangent helps calculate the angle opposite one of these sides.

• Use the tangent function: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)
• Here, the opposite side is the river's speed, and the adjacent is the swimmer's speed

The calculations are as follows:
\[\tan(\theta) = \frac{6}{2} = 3\]\[\theta = \tan^{-1}(3)\]
The angle \( \theta \) shows how much the swimmer's path deviates from a straight line perpendicular to the shore. This understanding is crucial when navigating real-life scenarios where multiple forces act simultaneously.