Problem 15 - Precalculus Mathematics for Calculus

Question

Express the given quantity in terms of the indicated variable. The distance (in mi) that a car travels in 45 min; \(s=\) speed of the car (in \(\mathrm{mi} / \mathrm{h}\) ).

Step-by-Step Solution

Answer

Distance is expressed as \(d = 0.75s\), where \(s\) is speed.

Step 1: Understanding the Problem

You are given a situation where a car travels for a certain time at a given speed, and you need to express the distance traveled by the car in terms of its speed. The way distance, speed, and time relate is by the formula \(d = s imes t\), where \(d\) is distance, \(s\) is speed, and \(t\) is time.

Step 2: Convert Time to Hours

The time given is 45 minutes, but our speed is in miles per hour, so we need to convert 45 minutes into hours because our units should be consistent. Since there are 60 minutes in an hour, we calculate the time in hours as \(45/60 = 0.75\) hours.

Step 3: Apply the Distance Formula

Now that we have the time in hours, use the formula for distance: \(d = s imes t\). Substitute \(t = 0.75\) hours into the equation: \(d = s imes 0.75\).

Step 4: Final Expression

The distance traveled is expressed as \(d = 0.75s\). This equation now represents the distance in terms of the speed \(s\) of the car.

Key Concepts

Distance Formula

In the realm of physics and everyday life, understanding how distance, speed, and time relate to each other is fundamental. The distance formula describes a straightforward relationship between these three elements. Simply put, the distance formula is given by \( d = s \times t \), where:

• \( d \) stands for the distance traveled.
• \( s \) represents the speed or velocity.
• \( t \) indicates the time duration over which the travel occurs.

This formula tells us that the distance is the product of speed and time. When we know any two of these variables, we can always find out the third. For instance, if a car travels at 60 miles per hour for 2 hours, it covers a distance of \( d = 60 \times 2 = 120 \) miles.

Unit Conversion

Unit conversion is an essential step when working with distance-speed-time problems. It ensures that all measurements align correctly, making calculations accurate and meaningful. In this exercise, the given time was in minutes, while speed was in miles per hour, necessitating a conversion to keep units consistent.To convert minutes into hours, remember that there are 60 minutes in an hour. Thus, you divide the number of minutes by 60.

• For 45 minutes, the conversion process is \( 45/60 = 0.75 \) hours.

By converting to hours, your problem aligns perfectly with the miles per hour speed unit, and you can proceed with solving the problem without discrepancies.Correct unit conversion is crucial, as failing to convert can lead to incorrect answers.

Equation Manipulation

Equation manipulation involves rearranging and simplifying equations to make them easier to work with. This process is essential in mathematics to derive useful relationships and solutions from known formulas. In our exercise, after converting the time to hours, we needed to express the distance in terms of the speed \(s\).Starting with the formula \( d = s \times t \), where \( t = 0.75 \) hours, we substitute \( t \) into the equation:

• \( d = s \times 0.75 \)

Simplifying this equation, we aim to isolate \( d \) by expressing it directly in terms of \( s \). This result, \( d = 0.75s \), succinctly represents the relationship between distance and speed, given the specific time frame of 45 minutes.Being adept at these manipulations enables you to swiftly solve various mathematical problems, especially when dealing with algebraic expressions.