Problem 26 - Elementary and Intermediate Algebra

Question

Substitute to find the value of each expression. Travel Time. The length of a flight from Seattle, Washington, to St. Paul, Minnesota, is approximately \(1400 \mathrm{mi} .\) The time, in hours, for the flight is $$ \frac{1400}{v} $$ where \(v\) is the velocity, in miles per hour. How long will a flight take at a velocity of \(400 \mathrm{mph} ?\)

Step-by-Step Solution

Answer

3.5 hours

Step 1: Identify given values

The distance of the flight from Seattle, Washington, to St. Paul, Minnesota is given as 1400 miles, and the velocity is given as 400 miles per hour.

Step 2: Write down the formula

The formula to find the time of the flight is given by \[ \text{Time} = \frac{1400}{v} \] where \(v\) is the velocity.

Step 3: Substitute the given velocity into the formula

Substitute \(v = 400 \) into the formula: \[ \text{Time} = \frac{1400}{400} \]

Step 4: Simplify the expression

Divide 1400 by 400: \[ \text{Time} = 3.5 \text{ hours} \]

Step 5: State the answer

The flight will take 3.5 hours at a velocity of 400 mph.

Key Concepts

substitution method

The substitution method is an algebraic technique where values are replaced into expressions or equations. It involves substituting a given variable with a specific value to simplify and solve equations. For instance, in our problem, we were given the formula to calculate travel time. The formula is: \ \ \[ \text{Time} = \frac{1400}{v} \] \ \ Here, we substituted the given velocity \(v = 400 \ \text{mph} \) into the formula. This helps us calculate how long a flight will take at a particular speed. This technique is particularly useful in a wide range of math problems, from simple arithmetic to more complex algebraic equations. Make sure to substitute accurately to avoid errors and always double-check your substitutions.

velocity and distance

Velocity and distance are fundamental concepts in understanding motion. \- **Velocity** is the speed of something in a given direction, measured as distance over time (miles per hour or kilometers per hour). \ - **Distance** is the total length of the path traveled.\ \ In this exercise, the distance is the length of the flight between Seattle and St. Paul, which is 1400 miles. The velocity is the speed of the flight given as 400 mph. These values are used in the formula: \ \[ \text{Time} = \frac{1400}{v} \] to determine the travel time. Understanding how velocity and distance relate helps in solving problems involving motion and travel.

simplifying fractions

Simplifying fractions is an important skill in algebra. It involves reducing a fraction to its simplest form. In our problem, the fraction we simplified was: \ \[ \frac{1400}{400} \] By dividing both the numerator (1400) and the denominator (400) by their greatest common divisor, we simplified the expression to 3.5. Simplification makes it easier to understand and interpret fractions. It involves: \ - Identifying a common factor \ - Dividing both the numerator and the denominator by this factor \ Keep practicing to improve your ability to simplify fractions quickly and accurately.

travel time calculation

Travel time calculation is essential for planning and understanding movements, especially over long distances. The general formula used is: \ \[ \text{Time} = \frac{\text{Distance}}{\text{Velocity}} \] \ By dividing the distance by the velocity, we get the time required for the travel. In our example, the distance is 1400 miles, and the velocity is 400 mph. Thus: \ \[ \text{Time} = \frac{1400}{400} = 3.5 \text{ hours} \] \ This formula is useful not just for flights but for any travel scenario involving a constant speed. It helps in estimating arrival times and planning efficient travel routes.