Problem 33 - Beginning and Intermediate Algebra

Question

If distance is held constant, the time it takes to travel that distance is inversely proportional to the speed at which one travels. If it takes 14 hr to travel the given distance at \(60 \mathrm{mph}\), how long would it take to travel the same distance at 70 mph?

Step-by-Step Solution

Answer

It would take 12 hours to travel the same distance at 70 mph.

Step 1: Find the constant of proportionality

Since we know that \(t \cdot s = k\), we can determine k using the given information that it takes 14 hours to travel the given distance at 60 mph: \[ k = t \cdot s = (14 \text{ hours}) \cdot (60 \mathrm{mph}) \] Calculating k we get: \[ k = 840 \, \text{hour}\cdot\text{mph} \]

Step 2: Use the constant of proportionality to find the time at 70 mph

Now that we know the constant of proportionality, k, we can use it to find the time it takes to travel the same distance at 70 mph: \[ t = \frac{k}{s} = \frac{840 \, \text{hour}\cdot\text{mph}}{70 \mathrm{mph}} \] Calculating the time t, we get: \[ t = 12 \text{ hours} \] Thus, it would take 12 hours to travel the same distance at 70 mph.

Key Concepts

Constant of Proportionality

When dealing with relationships where one quantity varies inversely with another, the constant of proportionality (often denoted by \( k \)) becomes a crucial element. In simple terms, inverse proportionality means that as one value increases, the other decreases. The constant \( k \) ensures that the product of the two varying elements remains unchanged. For example, if distance is fixed and speed increases, time must decrease to keep their product, which is the constant of proportionality, unchanged. In our exercise, the equation is given by \( t \cdot s = k \), where \( t \) is time and \( s \) is speed. By multiplying the known time (14 hours) by the speed (60 mph), we determine that the constant \( k \) is 840 hour\( \cdot \)mph. This constant helps in recalculating one variable when the other changes, without having to measure distance directly each time.

Distance-Speed-Time Relationship

The distance-speed-time formula is a fundamental concept in understanding motion. The relationship can be summed up by the formula \( \, d = s \cdot t \, \), where \( d \) represents distance, \( s \) stands for speed, and \( t \) denotes time. This formula helps determine one of the three variables when the other two are known. In this exercise, distance remains constant, highlighting an inverse proportionality between speed \( s \) and time \( t \). This means that as the speed increases, the time taken to cover the same distance decreases. Such a relationship is crucial in many real-life situations, like planning travel times or optimizing routes.

Algebraic Problem Solving

Algebraic problem solving involves using mathematical formulas and equations to find unknown values. Here, we use the equation \( \, t \cdot s = k \), an inverse relationship of time and speed, to solve for time at a different speed.To find how long it would take to travel the same distance at 70 mph, we rearrange the formula to solve for time \( t \), given the constant \( k \). By plugging in the values, we use \( \, t = \frac{k}{s} \, \) to determine that \( t = 12 \) hours. This type of problem-solving enables us to tackle various real-world scenarios where direct measurements aren't feasible. It allows us to make predictions or necessary adjustments based on changes in related variables.