Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

A boy's toy boat slips from his grasp at the edge of a straight river. The stream carries it along at 5 feet per second. A crosswind blows it toward the opposite bank at 4 feet per second. If the boy runs along the shore at 3 feet per second following his boat, how fast is the boat moving away from him when \(t=3\) seconds?

Step-by-Step Solution

Answer

4.47 feet per second.

Step 1: Define Directions and Components

The boat's motion can be broken into two components: along the river and across the river. The boat moves at 5 ft/s along the river, and the crosswind moves it at 4 ft/s across the river.

Step 2: Determine Initial Positions

Initially, at time \(t=0\), the boat is at the origin in our coordinate system starting from (0,0). At the same time, the boy is also standing at the origin, tracking the boat along the river's edge.

Step 3: Calculate Position at t=3 Seconds

After \(t=3\) seconds, the boat has traveled \(5 \times 3 = 15\) feet along the river and \(4 \times 3 = 12\) feet across the river. The boy has traveled \(3 \times 3 = 9\) feet along the river.

Step 4: Calculate Relative Position Vector

The position of the boat relative to the boy at \(t=3\) seconds is from the point \((9, 0)\) to the point \((15, 12)\).

Step 5: Use Pythagorean Theorem for Distance

The distance from the boy to the boat is \(\sqrt{(15 - 9)^2 + (12 - 0)^2} = \sqrt{6^2 + 12^2} = \sqrt{36 + 144} = \sqrt{180}\).

Step 6: Differentiate to Find Rate of Separation

Differentiating the distance \(d\) with respect to time \(t\) gives us the rate of separation. The rate is \(\frac{d}{dt} \sqrt{180} = \frac{d}{dt} (13.416) = 13.416\) feet per second. However, we must include the directional velocities to find the rate of separation between boat and boy.

Step 7: Calculate Resultant Separation Speed

The velocity in the river direction is \((5 - 3) = 2\) ft/s, and across the river is \(4\) ft/s. Thus, the resultant speed is \(\sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 4.47\) feet per second.

Key Concepts

Relative Motion

In our scenario, relative motion helps us understand how the boat and boy move in relation to each other. The boat moves because of two influences: the current and the wind. The boy moves along the shore.

• The current carries the boat downstream at 5 feet per second.
• A crosswind pushes it sideways at 4 feet per second.
• The boy follows along the shore at 3 feet per second.

To find how the boat moves away from the boy, their speeds and directions help determine how their positions change over time. Relative motion blends these movements to show how fast the boat gets away from the boy as time passes.
By understanding both the river's (5 ft/s) and boy's (3 ft/s) speeds, we can find how the boat gets further from the boy each second.

Pythagorean Theorem

The Pythagorean theorem is crucial for finding the straight-line distance between two points when you know the perpendicularly aligned distances. After the boat has traveled both downstream and across the river, its position forms a right triangle with the boy's position.
The principle is laid out by the equation: \[a^2 + b^2 = c^2\] where \(c\) is the hypotenuse—the direct line of the boat's location from the boy. Here,

• \(a\) is the difference in the downriver distance, \(6\) feet.
• \(b\) is the distance across the river, \(12\) feet.

By inserting these values into the equation, we discover:\[c = \sqrt{6^2 + 12^2} = \sqrt{36 + 144} = \sqrt{180}\] This value is the direct distance of the boat from the boy.

Differentiation

Differentiation is a powerful tool in calculus used to determine how quantities change over time, often in terms of speed or rate of change. Here, it is used to find out how fast the boat and boy are moving apart as time progresses.
From the right-angled triangle formed by their movements, the hypotenuse's length represents the separation distance. Differentiation helps calculate the rate at which this distance changes. Although initially, we computed the distance separation at a fixed time, differentiation uncovers the dynamic aspect, showing how the rate evolves.
After determining the positional changes through Pythagorean theorem, differentiating helps us specify that overall, the boat's separation speed is \(\sqrt{20} = 4.47\) feet per second.

Kinematics

Kinematics studies the motion of objects without considering the forces that cause the motion. In this problem, it helps us analyze both linear and lateral movements of objects.

• Linear: The boat's movement downstream is at a constant speed due to the river current.
• Lateral: The crosswind introduces a perpendicular speed that moves the boat sideways.
• The boy's consistent speed along the bank also illustrates linear motion.

By considering the simultaneous velocities from the river and the crosswind, and not the forces behind them, kinematics allows us to find how fast and in what direction the boat moves. This includes defining the resultant motion using both the river and lateral movements to derive the final separation speed between the boat and the boy.