Problem 74 - University Physics with Modern Physics

Question

A fence post is 52.0 m from where you are standing, in a direction 37.0\(^{\circ}\) north of east. A second fence post is due south from you. How far are you from the second post if the distance between the two posts is 68.0 m?

Step-by-Step Solution

Answer

The second post is approximately 59.13 m away from you.

Step 1: Define the Coordinate System

Set up a coordinate system with your current position at the origin, pointing east as the positive x-axis, and north as the positive y-axis. The first post is located at 52.0 m, 37.0° north of east.

Step 2: Determine the First Post's Coordinates

Calculate the x and y coordinates of the first post. Use \(x = 52.0 \times \cos(37.0^{\circ})\) and \(y = 52.0 \times \sin(37.0^{\circ})\).

Step 3: Calculate Coordinates

Compute the numerical values: \(x = 52.0 \times \cos(37.0^{\circ}) \approx 41.57\), \(y = 52.0 \times \sin(37.0^{\circ}) \approx 31.3\). The coordinates of the first post are approximately (41.57, 31.3).

Step 4: Translate Problem to Second Post

Since the second post is due south from you, it lies on the y-axis at coordinates (0, -b), where b is the distance from your position to the second fence post.

Step 5: Distance Between Posts Using Pythagorean Theorem

The positions of the posts form a right triangle, with sides made by distances on the x and y axes. Use the Pythagorean theorem: \(\sqrt{(41.57 - 0)^2 + (31.3 - (-b))^2} = 68.0\) to find b.

Step 6: Solve for b

Substitute and solve for the equation: \(\sqrt{41.57^2 + (31.3 + b)^2} = 68.0\). After squaring both sides and simplifying, solve for b.

Step 7: Rearrange and Simplify

Rearrange the equation to find \(b\approx \sqrt{68.0^2 - 41.57^2} - 31.3 \), then calculate to get \(b\approx 59.13 m\).

Step 8: Conclusion

The distance from you to the second post is approximately 59.13 m.

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in mathematics that relates the lengths of the sides of a right triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, it is expressed as: \[ c^2 = a^2 + b^2\] where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides of the triangle. This theorem is extremely useful in many areas of math and science, especially when dealing with problems involving distances and triangles. In the context of our fence post problem, we use the Pythagorean theorem to calculate the missing side length of a triangle formed by the two fence posts. By knowing the coordinates of the first post and the distance between the two posts, we can identify a right triangle where the hypotenuse is 68.0 meters. Using this equation, we find the length of the vertical side, which represents the distance from the starting point to the second fence post.

Coordinate system

A coordinate system is a mathematical tool used to determine positions in space by assigning coordinates to each point. The most common one is the Cartesian coordinate system, which uses two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point on this plane can be defined by a pair of numerical coordinates: \( (x, y) \).

• Origin: The starting point (0, 0) where the axes intersect.
• Positive x-axis: Extends to the right.
• Positive y-axis: Extends upwards.

For our example, setting up a coordinate system helps visualize and calculate distances between points. When considering the directional terms - 'north of east' - we position the first post at a specific angle (37°) from the east direction. This involves determining exact x and y distances using trigonometry, based on the angle and total distance from the origin (52.0 meters). The resulting coordinates (41.57, 31.3) accurately represent the post's position on this grid.

Trigonometric functions

Trigonometric functions like sine, cosine, and tangent, help translate angles and lengths in right triangles into algebraic terms. These functions are essential when dealing with problems that involve angles, such as our fence post example.

• Sine (\( \sin \)): Relates the opposite side to the hypotenuse: \( \sin(\theta) = \frac{opposite}{hypotenuse} \)
• Cosine (\( \cos \)): Relates the adjacent side to the hypotenuse: \( \cos(\theta) = \frac{adjacent}{hypotenuse} \)

In this problem, the angle given is 37° 'north of east'. To find the x (horizontal) and y (vertical) components of the distance to the first post, we use:\[x = 52.0 \times \cos(37.0^{\circ}) \approx 41.57\]\[y = 52.0 \times \sin(37.0^{\circ}) \approx 31.3\]These equations allow us to break down a complex direction into simple horizontal and vertical movements, enhancing our understanding of spatial relationships and distance calculations.