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Chapter 1: Problem 13

A highway is to be built between two towns, one of which lies 35.0 km south and 72.0 km west of the other. What is the shortest length of highway that can be built between the two towns, and at what angle would this highway be directed with respect to due west?

Short Answer

Expert verified
The shortest highway is about 80.1 km and is directed 26.0掳 south of due west.

Step by step solution

01

Determine the Distance Using the Pythagorean Theorem

To find the shortest highway distance between the two towns, we use the Pythagorean theorem. The situation describes a right triangle with legs of 35.0 km (south) and 72.0 km (west). The hypotenuse, which represents the highway, can be calculated using: \[ c = \sqrt{a^2 + b^2} \] where \( a = 35.0 \) km and \( b = 72.0 \) km. Substituting these values gives: \[ c = \sqrt{(35.0)^2 + (72.0)^2} \] \[ c = \sqrt{1225 + 5184} \] \[ c = \sqrt{6409} \] \[ c \approx 80.1 \] Thus, the shortest length of the highway is approximately 80.1 km.
02

Find the Angle With Respect to Due West

Next, we need to calculate the angle at which the highway is directed with respect to due west. We use trigonometry, specifically the tangent function, which relates the opposite and adjacent sides of a right triangle: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \] Here, the opposite side is the south direction (35.0 km), and the adjacent side is the west direction (72.0 km). Therefore: \[ \tan(\theta) = \frac{35.0}{72.0} \] \[ \theta = \tan^{-1}\left(\frac{35.0}{72.0}\right) \] Using a calculator, we find: \[ \theta \approx 26.0^\circ \] Thus, the highway should be directed approximately 26.0掳 south of due west.

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Key Concepts

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

Right Triangle
A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This angle helps create a relationship between the three sides of the triangle. In a right triangle:
  • The side opposite the right angle is called the hypotenuse.
  • The other two sides are known as the legs.
To understand how to find the shortest highway, consider that the towns' positions describe a right triangle. One town is 35 km south, and the other is 72 km west, forming the two legs of this triangle. The hypotenuse, or the longest side, will be the direct route鈥攐r the highway鈥攖hat we aim to calculate.
To solve the problem, we must find the length of the hypotenuse using the Pythagorean Theorem, which connects the lengths of the sides in every right triangle.
Trigonometry
Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. In particular, it is extremely useful for analyzing right triangles, which are fundamental in geometry problems.
In this exercise, trigonometry helps determine the angle at which the highway bends with respect to the westward direction. By applying trigonometric functions, we can solve real-world problems where distances and angles matter significantly.
In the given scenario, the relationship between the triangle's sides (south and west) and the highway's angle is key, which is where specific trigonometric functions like tangent come into play.
Tangent Function
The tangent function is one of the primary trigonometric functions, often written as \( \tan(\theta)\). This function relates two sides of a right triangle:
  • Opposite side (in this problem, 35 km south)
  • Adjacent side (in this problem, 72 km west)
Using the tangent function, we can calculate the angle at the junction of these sides, providing direction for the shortest path, here called the highway.
The formula \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \) allows you to find the angle \( \theta \) that gives precise directional guidance.
In our example, with the calculated tangent \( \theta = \tan^{-1}\left(\frac{35.0}{72.0}\right)\), we determine that this highway needs to be directed approximately 26.0掳 south of due west. This angle ensures the shortest and most direct route between the towns.

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Most popular questions from this chapter

Vector \(\overrightarrow{\mathbf{A}}\) has a magnitude of 12.3 units and points due west. Vector \(\overrightarrow{\mathbf{B}}\) points due north. (a) What is the magnitude of \(\overrightarrow{\mathbf{B}}\) if \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\) has a magnitude of 15.0 units? (b) What is the direction of \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\) relative to due west? (c) What is the magnitude of \(\overrightarrow{\mathbf{B}}\) if \(\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}\) has a magnitude of 15.0 units? (d) What is the direction of \(\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}\) relative to due west?

A student sees a newspaper ad for an apartment that has 1330 square feet \(\left(\mathrm{ft}^{2}\right)\) of floor space. How many square meters of area are there?

During takeoff, an airplane climbs with a speed of \(180 \mathrm{m} / \mathrm{s}\) at an angle of \(34^{\circ}\) above the horizontal. The speed and direction of the airplane constitute a vector quantity known as the velocity. The sun is shining directly overhead. How fast is the shadow of the plane moving along the ground? (That is, what is the magnitude of the horizontal component of the plane's velocity?)

Vector \(\overrightarrow{\mathbf{A}}\) points along the \(+y\) axis and has a magnitude of 100.0 units. Vector \(\overrightarrow{\mathbf{B}}\) points at an angle of \(60.0^{\circ}\) above the \(+x\) axis and has a magnitude of 200.0 units. Vector \(\overrightarrow{\mathbf{C}}\) points along the \(+x\) axis and has a magnitude of 150.0 units. Which vector has (a) the largest \(x\) component and (b) the largest y component?

You are driving into St. Louis, Missouri, and in the distance you see the famous Gateway to the West arch. This monument rises to a height of \(192 \mathrm{m}\). You estimate your line of sight with the top of the arch to be \(2.0^{\circ}\) above the horizontal. Approximately how far (in kilometers) are you from the base of the arch?
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