Problem 101 - Precalculus

Question

Find the distance traveled (arc length) of a point that moves with constant speed \(v\) along a circle in time \(t\). $$v=4.5 \mathrm{mi} / \mathrm{hr}, t=20 \mathrm{min}$$

Step-by-Step Solution

Answer

The distance traveled is 1.5 miles.

Step 1: Convert Time to Hours

The time provided is in minutes, but we need it in hours to use in the formula with speed in miles per hour. Convert the time from minutes to hours. Since there are 60 minutes in an hour, we convert by dividing by 60: \[ t = \frac{20}{60} = \frac{1}{3} \text{ hours} \]

Step 2: Use the Formula for Distance

The formula to calculate distance traveled (arc length) when moving with constant speed is: \[ \text{Distance} = \text{Speed} \times \text{Time} \]Substitute the known values into the formula: \[ \text{Distance} = 4.5 \text{ mi/hr} \times \frac{1}{3} \text{ hr} \]

Step 3: Calculate the Distance

Calculate the distance by performing the multiplication: \[ \text{Distance} = 4.5 \times \frac{1}{3} = 1.5 \text{ miles} \]

Key Concepts

Constant Speed

When we talk about constant speed, we mean that an object is moving in such a way that it does not speed up or slow down. The speed remains the same throughout the entire journey. This is a simple yet important concept. It helps us predict how far an object will travel over a given period of time. Constant speed is common in scenarios where the object does not face changing hills or wind. Think of driving a car on a highway without having to use your brakes or accelerate. The car moves steadily at the speed you've set. This is what constant speed looks like, allowing us to use simple formulas to calculate distances traveled.

Distance Formula

The distance formula is an essential tool for calculating how far an object moves over time. It works beautifully with constant speed because the rate of motion doesn't change. The formula is given by \[ \text{Distance} = \text{Speed} \times \text{Time} \]This equation shows you that to find out how much distance is covered, you simply multiply the speed, which is the rate of travel, by the time spent traveling.

• Speed: How fast the object is moving, usually given in units like miles per hour (mi/hr).
• Time: How long the object has been moving, often measured in hours.
• Distance: How far the object has traveled.

In our exercise, the point moves at 4.5 mi/hr for 20 minutes. By using the formula, we easily find the arc length or the distance traveled along the circle.

Time Conversion

Time conversion is an important skill when applying the distance formula. Often, time is given in a unit that needs converting, like minutes to hours.To convert time correctly, remember these key steps:

• Identify the current unit of time. For example, 20 minutes in our problem.
• Find out what unit you need. Here, we're working with miles per hour, so hours are needed.
• Use the conversion ratio. There are 60 minutes in an hour, so divide the minutes by 60.

For our problem, we have:\[ t = \frac{20}{60} = \frac{1}{3} \text{ hours} \]This process ensures you’re in the right unit to work seamlessly with the speed given in miles per hour.