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Chapter 1: Problem 26

Two bicyclists, starting at the same place, are riding toward the same campground by two different routes. One cyclist rides \(1080 \mathrm{~m}\) due east and then turns due north and travels another \(1430 \mathrm{~m}\) before reaching the campground. The second cyclist starts out by heading due north for \(1950 \mathrm{~m}\) and then turns and heads directly toward the campground. (a) At the turning point, how far is the second cyclist from the campground? (b) What direction (measured relative to due east) must the second cyclist head during the last part of the trip?

Short Answer

Expert verified
(a) Approximately 1714.92 m, (b) 31.0掳 relative to due east.

Step by step solution

01

Calculate the Total Distance for the First Cyclist

The first cyclist travels due east and then due north. To find the total distance to the campground from the starting point, we use the Pythagorean theorem. The eastward distance is \(1080\, \text{m}\) and the northward distance is \(1430\, \text{m}\). The total distance \(d\) is given by: \[d = \sqrt{(1080)^2 + (1430)^2} = \sqrt{1166400 + 2044900} = \sqrt{3211300}\]Calculating this gives, \[d \approx 1791.98\, \text{m}\]
02

Determine the Remaining Distance for the Second Cyclist

The second cyclist travels due north for \(1950\, \text{m}\) and then directly toward the campground. We need to find the remaining distance \(y\) to the campground.Using the northward distance from the first cyclist's path (\(1430\, \text{m}\)): \[y = 1791.98\, \text{m} - 1950\, \text{m} = \text{(negative, means the cyclist overshot the campground due north)}\]Instead, we calculate using Pythagoras:\[y = \sqrt{1791.98^2 - (1950 - 1430)^2} = \sqrt{3211300 - 270400} = \sqrt{2940900}\]Calculating this gives \[y \approx 1714.92\, \text{m}\]
03

Find the Direction for the Second Cyclist

The second cyclist must travel at an angle \(\theta\) relative to due east. Draw a right triangle where \(y = 1714.92\, \text{m}\) is the hypotenuse.The horizontal (eastward) leg is the remaining distance eastward:\(865\, \text{m}\) \((1950\, \text{m} - 1080\, \text{m})\).The angle \(\theta\) is:\[\theta = \arctan\left(\frac{1950 - 1430}{865}\right)\]\[\theta \approx \arctan(0.600) \approx 31.0^\circ\]
04

Compile the Results

Based on the calculations:(a) The second cyclist is approximately \(1714.92\, \text{m}\) from the campground at the turning point.(b) The direction the cyclist must head during the last part of the trip is approximately \(31.0^\circ\) relative to due east.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pythagorean Theorem
In this problem, the Pythagorean Theorem helps us determine the distance between two points in a right triangle. This theorem is essential for understanding distances in two-dimensional navigation, where most paths can be thought of as forming a right triangle. The theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
  • Formula: For a triangle with legs of lengths \(a\) and \(b\), and hypotenuse \(c\), the relationship is \( c^2 = a^2 + b^2 \).
  • In our exercise, the first cyclist's route forms a right triangle. The eastward journey of \(1080 \, \text{m}\) and northward journey of \(1430 \, \text{m}\) are the legs. The campground is the endpoint after the hypotenuse, calculated to be \(1791.98 \, \text{m}\).
  • This ability to determine distances is crucial in navigation tasks.
Trigonometry in Navigation
Trigonometry is the mathematical study of triangles, especially right triangles. It is extremely helpful in navigation to determine angles and directions. When you know some of the sides and angles, trigonometry provides precise navigation instructions.
  • In navigation, understanding your direction relative to geographic markers is important. Trigonometric functions like sine, cosine, and tangent describe these relationships.
  • In the problem, we calculate an angle for the second cyclist using the \(\arctan\) (inverse tangent) function. This function helps find the angle that the cyclist should travel.
  • The angle \( \theta \) relative to due east is computed using the difference in latitude and longitude distances, showing how trigonometry connects distances and directional angles.
  • This is a practical demonstration of trigonometry's role in determining navigation angles. The answer is around \(31.0^{\circ}\).
Right Triangle Solving
When presented with a right triangle problem, there are systematic ways to solve it. This typically involves the Pythagorean Theorem and trigonometric ratios.
  • Identifying the sides: Begin by labeling the triangle's sides as opposites, adjacents, or hypotenuses relative to the angle you're working with.
  • Using Pythagorean theorem and trig functions: If you know two sides, you can always find the third. If you have one angle and one side, use trigonometry to solve for other angles and sides.
  • In our exercise, this involved calculating the distance from a turning point to a campground, and also determining the cycling direction. Solving required both calculating distances (using the Pythagorean Theorem) and angles (using the inverse tangent function).
  • Step by step solving makes these problems manageable, transforming what initially seems complex into something solvable with simple mathematics.

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Most popular questions from this chapter

Your friend has slipped and fallen. To help her up, you pull with a force \(\overrightarrow{\mathbf{F}},\) as the rawing shows. The vertical component of this force is 130 newtons, and the horizontal omponent is 150 newtons. Find (a) the magnitude of \(\overrightarrow{\mathbf{F}}\) and (b) the angle \(\theta\).

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