Problem 47 - Algebra 1: Concepts and Skills

Question

A tsunami is a huge fast-moving series of water waves that can be caused by disturbances such as underwater earthquakes or volcanic explosions. If a tsunami is traveling at a speed of 500 miles per hour across the Pacific Ocean, how far has it gone in 15 minutes? HINT: Convert 15 minutes to hours.

Step-by-Step Solution

Answer

The tsunami has traveled 125 miles in 15 minutes.

Step 1: Convert minutes to hours

To convert minutes to hours, divide the number of minutes by 60, since there are 60 minutes in an hour. So, 15 minutes is equal to \( \frac{15}{60} \) hours, which simplifies to 0.25 hours.

Step 2: Use the formula for distance

The formula for distance when given speed and time is 'distance equals speed times time', or \( d = st \). Substituting the given values, \( d = 500 \times 0.25 \).

Step 3: Calculate the distance

Multiplying 500 by 0.25 equals 125. Therefore, the tsunami has traveled 125 miles in 15 minutes.

Key Concepts

Converting Minutes to Hours

When it comes to managing time in mathematical problems, understanding how to convert units is fundamental. One common conversion is turning minutes into hours. Since there are 60 minutes in one hour, to convert minutes to hours, you divide the number of minutes by 60. For instance, for a 15-minute duration, the conversion to hours is given by the formula:
\[ \text{hours} = \frac{\text{minutes}}{60} \]
So,
\[ \frac{15}{60} = 0.25 \text{ hours} \]
This conversion is crucial in many real-world situations, such as calculating travel times or planning events. It serves as a basis for further calculations in various problems dealing with rates, such as speed.

Distance Speed Time Formula

A central concept in motion problems is the relationship between distance, speed, and time. This is encapsulated in a simple yet powerful formula:
\[ d = s \times t \]
where \(d\) stands for distance, \(s\) for speed, and \(t\) for time. With this formula, if you know any two of the variables, you can find the third. In the context of our tsunami problem, if the speed (\(s\)) is given as 500 miles per hour and the time (\(t\)) has been converted to 0.25 hours, you'd calculate the distance (\(d\)) using the formula:
\[ d = 500 \text{ mph} \times 0.25 \text{ hours} = 125 \text{ miles} \]
Real-world applications for this formula are vast, ranging from planning road trips to estimating the time of arrival for flights.

Arithmetic Problem-Solving

Problem-solving in arithmetic involves a series of logical steps to arrive at a solution. Key to this process is understanding the question, identifying the relevant information, choosing the correct arithmetic operations, and carrying out the calculations accurately. In the tsunami example, the problem-solving steps are clear: convert minutes to hours for the time, identify the given speed, and then apply the distance formula.
Effective problem-solving often requires that these steps are performed in a sequence, ensuring each operation is valid and correctly executed. Critical thinking aids in verifying the results, providing assurance that the solution makes sense in the context of the problem.

Real-World Algebra Applications

Algebra isn't just a subject learned in school—it's a tool used to solve a variety of real-world problems. Whenever we talk about unknown quantities and the relationships between them, algebra comes into play. For instance, determining the distance a tsunami travels over a period is a real-world application of algebra. By using variables and formulas, we translate a physical situation into a mathematical model that we can solve.
Understanding how to form equations based on real-life scenarios allows us to predict outcomes and make informed decisions. Whether it's budgeting, construction, or scientific research, algebra forms the backbone of problem-solving in numerous fields. By grasping these aspects, students can appreciate the value of algebra beyond the classroom.