Problem 37 - Algebra 1: Concepts and Skills

Question

Find the distance traveled using \(d=r t\). A racecar driver goes at a speed of 170 miles per hour for 2 hours.

Step-by-Step Solution

Answer

The racecar driver has traveled 340 miles.

Step 1: Identify the Given Values

From the problem, we identify the values that have been provided. The driver's speed or rate, denoted as \(r\), is 170 miles per hour. The time, denoted by \(t\), that the driver traveled for is 2 hours.

Step 2: Substitute the Values into the Formula

Now that we have our values, we plug them into the distance formula: \(d = r t\). Substituting the values, we get \(d = 170 \times 2\).

Step 3: Calculate the Distance

Now we perform the multiplication, it gives us a result of \(d = 340\) miles. This is the total distance the racecar driver has traveled.

Key Concepts

Understanding Rate

The concept of "rate" in mathematics and physics is fundamental. It's often used to describe how fast something happens or how much of something occurs within a certain period of time. Rate is essentially a ratio between two quantities, with one quantity divided by another. In this exercise, the rate refers to the speed of the racecar. When we say the racecar travels at a rate of 170 miles per hour, it means that every hour, the car covers 170 miles. In formulas, rate is usually represented by the letter 'r'. It's important to understand rates because they let us compare different types of values, such as distance with time in this case. Key features of rates:

• They always involve two different units (e.g. miles per hour, dollars per day).
• They show how one quantity changes in relation to another.
• They can be used in a variety of real-world contexts, such as speed, density, or population growth.

Understanding rate helps solve real-world problems quickly and efficiently.

Importance of Time

Time is a measure we use every day to evaluate how long something takes. In the context of the distance formula, time determines how long an object, such as a racecar, is moving at a given rate. In the formula used in this exercise, time is denoted as 't'. Here, the problem gives us the time the racecar is in motion, which is 2 hours. Time is a crucial variable because without it, we cannot calculate distances or speeds accurately. Some essential properties of time include:

• It's continuous and always moves forward.
• It's usually measured in seconds, minutes, hours, etc.
• It helps us synchronize activities and understand sequences of events.

Understanding how time affects movement is key to making accurate predictions and calculations in both everyday life and scientific contexts.

Multiplication in Action

Multiplication is a basic yet powerful mathematical operation essential for calculating products of numbers. In the distance formula, multiplication is critical in combining rate and time to find the distance. The operation symbolized by '×' helps us express the relationship between these two quantities. In the original exercise, the multiplication of 170 (rate) and 2 (time) results in 340, which represents the distance traveled. Let’s look at some notable points about multiplication:

• It's a repeated addition; in this context, adding 170 miles per hour twice since the car travels for 2 hours.
• It's commutative and associative, meaning that the order or grouping of numbers doesn't change the result.
• It's fundamental across many mathematical applications beyond just the distance formula, like area calculation or scaling models.

Multiplication links different aspects of a problem together, and mastering it will allow you to handle more complex mathematical situations with ease.