Problem 15
Question
CAR TRIP Suppose you drive 630 miles from St. Louis, Missouri, to Dallas, Texas, in 10.5 hours. Solve the equation \(630=r(10.5)\) for \(r\) to find your average speed.
Step-by-Step Solution
Verified Answer
The average speed of the trip from St. Louis, Missouri to Dallas, Texas is 60 miles per hour.
1Step 1: Understand the equation
The equation represents the relationship between distance, speed, and time. In the equation \(630 = r(10.5)\), 630 is the distance covered, 10.5 is the time spent, and \(r\) represents the average speed which is the quantity we need to find.
2Step 2: Rearrange the equation
To isolate \(r\), divide both sides of the equation by 10.5. The equation now becomes \(r = \frac{630}{10.5}\).\}
3Step 3: Calculate the average speed
By performing the division \(\frac{630}{10.5}\), the average speed \(r\) is determined to be 60 miles per hour.
Key Concepts
Solving EquationsDistance Speed Time RelationshipAlgebraic ExpressionsDistance Formula
Solving Equations
When you come across an equation like the one in the car trip problem, solving it involves finding the value of the unknown variable, which in this case is the average speed, denoted by 'r'. The general strategy is to manipulate the equation so that the unknown is by itself on one side of the equal sign. For the exercise, we start with the equation
\(630 = r(10.5)\).
To isolate 'r', we perform the operation of division on both sides by 10.5, yielding:
\(r = \frac{630}{10.5}\).
This process of isolating the variable is fundamental to solving equations, and doing so allows us to directly calculate the value of 'r'.
Always remember to perform the same operation on both sides of an equation to maintain its balance!
\(630 = r(10.5)\).
To isolate 'r', we perform the operation of division on both sides by 10.5, yielding:
\(r = \frac{630}{10.5}\).
This process of isolating the variable is fundamental to solving equations, and doing so allows us to directly calculate the value of 'r'.
Always remember to perform the same operation on both sides of an equation to maintain its balance!
Distance Speed Time Relationship
The relationship between distance, speed, and time is a fundamental concept in physics and is given by the formula:
\(\text{Distance} = \text{Speed} \times \text{Time}\).
In our exercise, you can see this relationship in action. The distance traveled is 630 miles, and the time spent traveling is 10.5 hours. To find the average speed, you divide the distance by the time, as demonstrated in the solution. It's crucial to understand that if you know any two of the three variables, you can always find the third using this simple but powerful relationship.
\(\text{Distance} = \text{Speed} \times \text{Time}\).
In our exercise, you can see this relationship in action. The distance traveled is 630 miles, and the time spent traveling is 10.5 hours. To find the average speed, you divide the distance by the time, as demonstrated in the solution. It's crucial to understand that if you know any two of the three variables, you can always find the third using this simple but powerful relationship.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. In the average speed calculation, \(630 = r(10.5)\) is an algebraic expression that embodies the relationship between distance, speed, and time.
When we rearranged the equation by dividing both sides by 10.5, we transformed it into a simpler expression \(r = \frac{630}{10.5}\) that made it easier to solve for 'r'. Learning to work with algebraic expressions is critical for problem-solving in algebra because it allows you to describe relationships and perform calculations with different variables.
When we rearranged the equation by dividing both sides by 10.5, we transformed it into a simpler expression \(r = \frac{630}{10.5}\) that made it easier to solve for 'r'. Learning to work with algebraic expressions is critical for problem-solving in algebra because it allows you to describe relationships and perform calculations with different variables.
Distance Formula
The distance formula in the context of speed and time is essentially the equation
\(\text{Distance} = \text{Speed} \times \text{Time}\),
which we've used in our car trip problem. This formula is the cornerstone for calculating the distance traveled when the speed and time are known. Conversely, as in our problem, if the distance and time are known, we can rearrange the formula to solve for speed. The use of the distance formula transcends just travel problems; it's a vital tool in various fields like physics, navigation, and even sports analytics. Understanding how to apply and manipulate the distance formula is key to mastering the concept of motion.
\(\text{Distance} = \text{Speed} \times \text{Time}\),
which we've used in our car trip problem. This formula is the cornerstone for calculating the distance traveled when the speed and time are known. Conversely, as in our problem, if the distance and time are known, we can rearrange the formula to solve for speed. The use of the distance formula transcends just travel problems; it's a vital tool in various fields like physics, navigation, and even sports analytics. Understanding how to apply and manipulate the distance formula is key to mastering the concept of motion.
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