Problem 15 - Algebra 2 and Trigonometry

Question

When on vacation, the Ross family always travels the same number of miles each day. a. Does the time that they travel each day vary inversely as the rate at which they travel? b. On the first day the Ross family travels for 3 hours at an average rate of 60 miles per hour and on the second day they travel for 4 hours. What was their average rate of speed on the second day?

Step-by-Step Solution

Answer

a) Yes, time and rate vary inversely. b) Their average speed on the second day was 45 mph.

Step 1: Understand Inverse Variation

To determine if one quantity varies inversely with another, we need to check if the product of the two quantities is constant. In this context, if the time varies inversely with the rate, then \( \,\text{time} \times \text{rate} = \text{constant} \, \) must hold.

Step 2: Analyze Day 1 Travel

For the first day, the family travels for 3 hours at 60 miles per hour. The distance they travel can be calculated using the formula \( \, ext{distance} = ext{rate} \times ext{time} \, \). Thus, \( \, 3 \times 60 = 180 \, \) miles. This is the constant distance they travel each day.

Step 3: Determine Constant Value

From Step 2, we found the distance to be 180 miles. On subsequent days, if the time and rate truly vary inversely, the product \( \, ext{time} \times ext{rate} \, \) should also equal 180 miles.

Step 4: Analyze Day 2 Travel

On the second day, the family travels for 4 hours. Using the inverse variation property (from Step 3), their average rate must satisfy \( \, 4 \times ext{rate} = 180 \, \). To find the rate, solve for it: \( \, ext{rate} = \frac{180}{4} = 45 \, \) miles per hour.

Step 5: Confirm Inverse Variation

Since the product for both days equals the same constant (180 miles), we can confirm that the time of travel varies inversely with their speed on each given day.

Key Concepts

Time and Rate Relationship

When discussing travel, the time and rate at which you move can change depending on various factors. In our scenario, time and rate have an inverse relationship. If one increases, the other decreases, maintaining a balance. This means if you increase your speed, the time taken is shorter, assuming the distance remains the same.

To determine this relationship concretely, you must check if the product of time and rate remains constant. This situation is described by the formula:

• \( \text{time} \times \text{rate} = \text{constant} \)

If this condition is met, you've found an inverse variation, where a change in one value inversely affects the other.

Calculating Speed

Calculating speed involves dividing distance by time. It's a simple yet powerful calculation. In the scenario with the Ross family, the task was to determine their speed on different days. By understanding that time and rate are inversely related, you can easily find the missing value if you know the others.

For example, knowing they travel 180 miles each day, you use the formula:

• \( \text{speed} = \frac{\text{distance}}{\text{time}} \)

On the first day, they traveled for 3 hours. Therefore,

• First Day: \( \frac{180}{3} = 60 \text{ miles per hour} \)
• Second Day: Given they traveled for 4 hours, you calculate \( \frac{180}{4} = 45 \text{ miles per hour} \)

This simple method provides a clear insight into their travel dynamics each day.

Constant Distance

Throughout their journey, the Ross family adheres to a constant travel distance each day. A constant distance means that no matter how fast or slow they travel, the total miles covered remains unchanged.

In their case, this set value is 180 miles daily. Maintaining a constant distance allows the comparison of time and rate to be straightforward while preserving consistency in their travel goals.

• Day 1: 3 hours at 60 mph results in 180 miles
• Day 2: 4 hours at 45 mph also results in 180 miles

This constancy helps us understand the constraints and allow the application of the inverse variation relationship effectively.

Algebraic Problem Solving

Algebraic problem solving helps you uncover relationships between variables. By setting up equations that represent real-world scenarios, complex problems become manageable.

For the Ross family's travel, we use algebra to find how rate and time vary inversely. Start with the rule \( \text{time} \times \text{rate} = \text{constant} \). By substituting known values, like time or distance, you solve for unknowns.

Consider the second day's travel where:

• \( 4 \times \text{rate} = 180 \)
• Divide both sides by 4: \( \text{rate} = \frac{180}{4} = 45 \text{ mph} \)

This simple algebraic manipulation shows the power of equations in deducing pivotal aspects of travel.