Problem 93 - Algebra for College Students

Question

Solve each problem. Driving speed. Jeremy drives 500 miles in \(x\) hours. Find a rational function \(S(x)\) that gives his average speed in miles per hour.

Step-by-Step Solution

Answer

\(S(x) = \frac{500}{x}\)

Step 1: Understand the Definition of Average Speed

Average speed is defined as the total distance traveled divided by the total time taken. For Jeremy, the total distance is 500 miles and the time taken is given as \(x\) hours.

Step 2: Write the Formula for Average Speed

The formula for average speed \(S(x)\) can be written as: \[S(x) = \frac{\text{Total Distance}}{\text{Total Time}}\]

Step 3: Substitute the Given Values

Substitute the given values into the formula. The total distance is 500 miles and the total time is \(x\) hours:\[S(x) = \frac{500}{x}\]

Key Concepts

Rational Functions

Rational functions are mathematical expressions where you have a ratio or fraction of two polynomials. In simpler terms, it’s like dividing one number by another, except we're using polynomial expressions.

For example, if we take a rational function like \(\frac{500}{x}\), it conveniently describes a relationship where 500 (a constant) is divided by a variable, in this case, x.

Rational functions are useful in lots of real-world scenarios. In Jeremy's case, it helps us calculate his average speed based on the distance traveled and the time taken.

When dealing with rational functions, always make sure you understand the meaning of each part of the expression. Here:

• 500 represents a constant distance in miles
• x represents the time taken in hours

Hence, \(\frac{500}{x}\) forms a rational function to give us the average speed. Always be careful to identify situations where the denominator (in this example x) is zero, as division by zero is undefined in mathematics.

Distance and Time Relationship

Distance and time are key components when calculating average speed. Understanding the relationship between distance, time, and speed is crucial.

The basic formula to remember here is: \(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\)

Here, distance is how far you've traveled and time is how long it took you to get there. Therefore, to find Jeremy's speed, we use the distance (500 miles) and the time (x hours) he takes.

This relationship can be applied in many practical scenarios:

• If you know the speed and the time, you can find the distance: \(\text{Distance} = \text{Speed} \times \text{Time}\)
• If you know the speed and distance, you can find time: \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\)

By organizing these relationships into algebraic expressions, you can solve many problems involving distance, time, and speed.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition, subtraction, multiplication, and division). In our case, the expression \(\frac{500}{x}\) is an algebraic expression that describes a situation.

When working with algebraic expressions, keep these in mind:

• Variables: These are the letters (e.g., x) that stand in for unknown values that can change.
• Constants: These are fixed values (e.g., 500).
• Operators: These include the addition, subtraction, multiplication, and division symbols.

Combining these correctly allows you to form meaningful expressions, equations, and ultimately solve problems.

To find the average speed, our algebraic expression \(\frac{500}{x}\) represents dividing a constant (500 miles) by a variable (x hours). This calculation helps us understand Jeremy’s performance, as it converts distance per time into speed.