Problem 15 - Algebra and Trigonometry

Question

\(7-18 \cdot\) 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

The distance traveled is \( 0.75s \) miles.

Step 1: Convert Time Units

The car travels for 45 minutes. Since the speed is given in miles per hour, convert the travel time from minutes to hours. We know that there are 60 minutes in an hour, thus 45 minutes can be converted to hours by dividing by 60: \( t = \frac{45}{60} = 0.75 \) hours.

Step 2: Use the Distance Formula

The formula to calculate distance is: \( \text{Distance} = \text{Speed} \times \text{Time} \). In this case, substitute the given values into the formula. The speed of the car is \( s \) and the time calculated before is 0.75 hours, thus: \( \text{Distance} = s \times 0.75 \).

Step 3: Express Distance in Terms of s

When expressing the distance traveled in 45 minutes in terms of the speed \( s \), we get \( \text{Distance} = 0.75s \).

Key Concepts

Time Conversion

To solve distance problems effectively, it's often necessary to convert units of time. In this particular exercise, the travel time was initially given in minutes. However, since the speed was measured in miles per hour (mi/h), we needed to convert those minutes into hours.

• To convert minutes to hours, remember that there are 60 minutes in one hour. So, divide the number of minutes by 60 to change them to hours.
• For example, 45 minutes is converted into hours by calculating \( \frac{45}{60} = 0.75 \) hours. This gives us a compatible unit for the speed measure in mi/h.

This conversion is crucial for accurately using the speed given. Always ensure your time unit matches the unit of speed when calculating distances.

Speed and Distance Relationship

Understanding how speed, time, and distance interact is key to solving many travel-related problems. These three components share a simple, direct relationship often encapsulated in the distance formula:

• The distance a vehicle travels is equal to its speed multiplied by the time it travels: \( \text{Distance} = \text{Speed} \times \text{Time} \).
• In scenarios where the speed is constant, as given by \( s \) mi/h in our problem, the distance can fluctuate based on the time traveled.

This relationship helps us solve for any one variable if we know the other two. In this situation, because the speed is constant, and we converted time to 0.75 hours, the distance traveled by the car is simply \( 0.75 \times s \). This formula is a handy tool for calculating trips given either the predicted speed or time.

Algebraic Expression

An algebraic expression allows us to represent distances concisely in terms of variables. In this exercise, we approached the solution by taking the formula for distance and expressing it mathematically in terms of the variable speed, \( s \).

• Once the correct units for time were established (0.75 hours), the next step was to plug these into the distance formula, \( \text{Distance} = s \times 0.75 \).
• This yields the expression \( 0.75s \), showing how the distance changes directly with different speeds.

By expressing travel distance as an algebraic expression, it not only simplifies calculations but also shows the direct dependence of distance on speed. You can now easily plug in various speeds to see how far you would travel in 45 minutes.