Problem 42 - Algebra and Trigonometry

Question

A car travels along a straight road, heading east for 1 h, then traveling for 30 min on another road that leads northeast. If the car has maintained a constant speed of 40 mi/h, how far is it from its starting position?

Step-by-Step Solution

Answer

The car is approximately 55.28 miles from the starting position.

Step 1: Calculate Distance for First Road

The car travels east for 1 hour at a speed of 40 mi/h. The distance traveled is given by the formula \( \text{distance} = \text{speed} \times \text{time} \). Therefore, the distance traveled east is \( 40 \text{ mi/h} \times 1 \text{ h} = 40 \text{ miles} \).

Step 2: Calculate Distance for Second Road

The car then travels northeast for 30 minutes (which is 0.5 hours) at the same speed. Using the formula \( \text{distance} = \text{speed} \times \text{time} \), the distance covered is \( 40 \text{ mi/h} \times 0.5 \text{ h} = 20 \text{ miles} \).

Step 3: Decompose Northeast Travel into Components

Since northeast is at a 45-degree angle to both north and east, the northeast distance can be split into two equal components: eastward and northward. Each component is \( \frac{20}{\sqrt{2}} = 10\sqrt{2} \) miles, because these components can be found using \( \text{Distance} \times \cos(45^\circ) \) or \( \text{Distance} \times \sin(45^\circ) \).

Step 4: Calculate Total Eastward Distance

The total distance traveled east is the sum of the initial eastward distance (40 miles) and the eastward component of the northeast travel \( 10\sqrt{2} \) miles. Thus, \( \text{Total Eastward Distance} = 40 + 10\sqrt{2} \).

Step 5: Total Northward Distance

The northward component of the northeast road is \( 10\sqrt{2} \) miles because there was no initial northward travel.

Step 6: Calculate Straight Line Distance from Start

Using the Pythagorean theorem, the straight-line distance from the starting position is: \( d = \sqrt{(40 + 10\sqrt{2})^2 + (10\sqrt{2})^2} \). Simplify this to find the distance.

Key Concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in trigonometry and is primarily used to relate the lengths of the sides of a right triangle. According to the theorem, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be expressed with the formula:\[ c^2 = a^2 + b^2 \]where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.
In the context of the car travel problem, we use the Pythagorean theorem to find out how far the car is from its starting position after traveling east and then northeast. By breaking down the northeast journey into two perpendicular components, we effectively create a right triangle, allowing us to calculate the direct distance using this well-known formula.

Vector Components

Vector decomposition is a crucial concept when dealing with movements at angles, such as traveling northeast. When something moves at an angle, it can be broken down into horizontal (east-west) and vertical (north-south) components to simplify calculations.
In the case of traveling northeast at a 45-degree angle, you split the total distance into two equal parts using trigonometric functions:

• Use \( \cos(45^\circ) \) for the eastward component.
• Use \( \sin(45^\circ) \) for the northward component.

For this exercise, the northeast travel of 20 miles had components of equal length. The calculations involved finding these components by multiplying 20 miles by \( \cos(45^\circ) \) or \( \sin(45^\circ) \), each yielding \( 10\sqrt{2} \) miles. This breakdown into vector components helps in accurately determining total movement in particular directions.

Distance Formula

The distance formula is often used to determine how far apart two points are in a plane. It's directly derived from the Pythagorean theorem. The formula is:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
For our problem, the distance formula is adapted to find how far the car is from its start point by substituting the total eastward and northward distances to compute the straight-line distance. In simpler terms, it’s assessing how straight you’ve "traveled" from corner-to-corner of the triangle formed by your journey.

Angle Decomposition

Angle decomposition is an invaluable technique in trigonometry, particularly in navigational tasks where exact movements are crucial. It involves breaking apart an angle into simpler components to facilitate easier calculations, often using sine and cosine.
For the exercise at hand, the travel directed northeast implies a 45-degree angle, perfectly placed between the coordinate axes. By decomposing this angle, we could easily translate a diagonal movement into clear northward and eastward paths using the trigonometric identities:

• \( \cos(45^\circ) = \frac{1}{\sqrt{2}} \)
• \( \sin(45^\circ) = \frac{1}{\sqrt{2}} \)

This breakdown allows us to simplify the problem and methodically use trigonometry to pinpoint exactly how far and in which directions the car has moved. It’s all about transforming a complex motion into manageable steps.