Problem 57 - Physics for Scientists and Engineers with Modern Physics

Question

Jean camps beside a wide river and wonders how wide it is She spots a large rock on the bank directly across from her. She then walks upstream until she judges that the angle between her and the rock, which she can still sce clearly, is now at an angle of \(30^{\circ}\) downstream (Fig. 16). Jean measures her stride to be about 1 yard long. The distance back to her camp is 120 strides. About how yards and in meters, yards and in meters, is the river?

Step-by-Step Solution

Answer

The river is about 69.24 yards or 63.25 meters wide.

Step 1: Understand the Situation

Jean is trying to calculate the width of the river, which can be visualized as a right triangle where one angle is known. She walks directly away from the point on the river bank opposite her starting point until she sees this point at a 30° angle from her new position downstream.

Step 2: Identify the Triangle Setup

The setup involves a right triangle where the width of the river is one side opposite the 30° angle, the path Jean walked is the adjacent side, and the line from Jean's new position to the rock is the hypotenuse.

Step 3: Convert Strides to Yards

Jean has taken 120 strides, and since her stride length is 1 yard, she has walked 120 yards along the river bank.

Step 4: Use Trigonometry to Set Up an Equation

In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. Since Jean observes the rock at a 30° angle, use the formula: \[tan(30^{\circ}) = \frac{\text{width of the river}}{120}\]

Step 5: Calculate the Width of the River in Yards

Using the known value for \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \), rearrange the equation as follows: \[\text{width of the river} = 120 \times \tan(30^{\circ})\]Substitute the value of \( \tan(30^{\circ}) \): \[\text{width of the river} = 120 \times \frac{1}{\sqrt{3}}\]\[\text{width of the river} \approx 120 \times 0.577\]\[\text{width of the river} \approx 69.24 \text{ yards}\]

Step 6: Convert the Measurement to Meters

To convert the result from yards to meters, use the conversion factor where 1 yard is approximately 0.9144 meters.\[\text{width of the river in meters} = 69.24 \times 0.9144\]\[\text{width of the river in meters} \approx 63.25\text{ m}\]

Key Concepts

Right Triangle

A right triangle is a special type of triangle where one of the angles is exactly 90 degrees. It consists of three sides: the hypotenuse, which is the longest side opposite the right angle, and two other sides, referred to as the adjacent and opposite sides, relative to a specific angle in the triangle. To solve many trigonometric problems, identifying the right triangle is crucial, as it allows the application of trigonometric functions like sine, cosine, and tangent.

In our exercise with Jean, the right triangle is formed by the following:

• The river's width: the side opposite the 30-degree angle.
• The distance Jean walked: the adjacent side.
• The line from Jean's new position to the rock: the hypotenuse.

Recognizing this setup helps us utilize mathematical principles effectively to find unknown dimensions, such as the width of the river.

Angle of Depression

The angle of depression is the angle formed by the line of sight from an observer looking downward to an object below. It is measured from the horizontal line, downward to the object being surveyed. In essence, it is the down angle observed from the horizontal.

In the context of this problem:

• Jean's line of sight to the rock forms the angle of depression.
• Even though it's termed 'depression', for problem-solving, it equates to the horizontal angle from her line of sight upstream to the rock downstream.

Understanding the angle of depression helps in setting up the right geometry for using trigonometric functions to calculate distances across horizontal and vertical planes.

Tangent Function

The tangent function is one of the primary trigonometric functions. It is defined as the ratio of the opposite side to the adjacent side in a right triangle. This function is highly useful when dealing with angles and lengths where direct measurement is not feasible.

With Jean's problem, we use the tangent function as follows:

• The angle is 30 degrees.
• The opposite side is the river's width.
• The adjacent side is the 120 yards Jean walked.

The formula is then set up as: \[ \tan(30^{\circ}) = \frac{\text{width of the river}}{120} \]Solving this provides the river's width, utilizing the known tangent value for a 30-degree angle: \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \), which is approximately 0.577.

Unit Conversion

Unit conversion involves changing the measurement units of a quantity to different units, maintaining the same quantity's actual size. This is essential in mathematics and science, where problems often involve using different unit systems.

In this exercise, we need to convert the river's width from yards to meters:

• We first calculate the width in yards, which is approximately 69.24 yards.
• The conversion factor from yards to meters is 1 yard = 0.9144 meters.

The calculation becomes:\[ \text{Width in meters} = 69.24 \times 0.9144 \]This conversion results in a width of approximately 63.25 meters. Understanding how to effectively use unit conversion is key in ensuring results are accurate and in the desired units of measure.