Problem 41 - Algebra and Trigonometry

Question

Two straight roads diverge at an angle of \(65^{\circ} .\) Two cars leave the intersection at \(2 : 00\) P.M. one traveling at 50 \(\mathrm{mi} / \mathrm{h}\) and the other at 30 \(\mathrm{mi} / \mathrm{h}\) . How far apart are the cars at \(2 : 30 \mathrm{P.M.?}\)

Step-by-Step Solution

Answer

The cars are approximately 23.08 miles apart at 2:30 PM.

Step 1: Understand the Problem

We have two roads diverging at an angle of \(65^{\circ}\). Two cars start from the intersection at the same time, one traveling at \(50\, \mathrm{mi/h}\) and the other at \(30\, \mathrm{mi/h}\). We need to find the distance between the two cars after 30 minutes.

Step 2: Calculate Distance Traveled by Each Car

First, calculate the distance each car travels in 30 minutes. Since 30 minutes is 0.5 hours:For the first car: \[\text{Distance}_1 = 50\, \mathrm{mi/h} \times 0.5\, \mathrm{h} = 25\, \mathrm{mi}\]For the second car: \[\text{Distance}_2 = 30\, \mathrm{mi/h} \times 0.5\, \mathrm{h} = 15\, \mathrm{mi}\]

Step 3: Apply the Law of Cosines

To find the distance between the two cars, we apply the Law of Cosines in the triangle formed by their paths:\[ c^2 = a^2 + b^2 - 2ab\cos(C) \]where \(a = 25\, \mathrm{mi}, b = 15\, \mathrm{mi}, C = 65^{\circ}\).Plugging in the values:\[ c^2 = 25^2 + 15^2 - 2 \times 25 \times 15 \times \cos(65^{\circ}) \]

Step 4: Solve for Distance Between the Cars

Calculate \(\cos(65^{\circ})\) and then solve for \(c\):\[\cos(65^{\circ}) \approx 0.4226\]\[ c^2 = 625 + 225 - 2 \times 25 \times 15 \times 0.4226 \]Calculate further:\[ c^2 = 850 - 317.45 \]\[ c^2 = 532.55 \]\[ c = \sqrt{532.55} \approx 23.08 \text{ miles}\]

Step 5: Provide the Final Answer

The distance between the two cars at \(2:30\, \, \mathrm{P.M.}\) is approximately \(23.08\, \mathrm{miles}\).

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that explores the relationships between the sides and angles of triangles. It plays a vital role in various fields, including physics, engineering, and navigation. In this exercise, trigonometry is utilized to determine the distance between two cars using the Law of Cosines. This law is particularly important for solving problems where you know two sides and the included angle, as is the case when you deal with diverging paths and need to calculate separation based on angular differences. Knowing how angles relate to distances helps solve complex real-world problems with ease.

Distance Calculation

Calculating distance accurately is crucial in both everyday life and scientific contexts. In this problem, the first step was to calculate how far each car traveled in 30 minutes. Given one car travels at 50 miles per hour and the other at 30 miles per hour, converting the time to a fraction of an hour allows us to measure their progress. It's a simple multiplication:

• For the car traveling at 50 miles/hour: 50 mi/h * 0.5 h = 25 miles
• For the car traveling at 30 miles/hour: 30 mi/h * 0.5 h = 15 miles

Using these distances, we can then apply more advanced mathematical techniques to find their separation.

Angular Separation

Angular separation refers to the angle at which two paths diverge from a common point, akin to how celestial objects are positioned along the sky. In practical terms, it affects the calculation of distances when paths are not parallel. In our scenario, a crucial component was understanding the angle of 65 degrees between the roads on which the cars travel. This angle affects how far apart the cars will be after both travel a certain distance. By using the Law of Cosines, which adjusts for angular separation, we can calculate the "c" side of a triangle, thus finding the real-world separation in miles between the two cars.