Solving Speed Problems: A Real-Life Application in Algebra

Algebra is more than just a theoretical concept—its principles are crucial in solving real-life problems, such as those involving speed and distance. One classic example involves determining the speed at which a person drives 180 miles under certain conditions. Let's dive into the problem and see how algebraic equations can help find the solution.

Problem Statement:

You drove 180 miles at a constant rate of speed. If you had driven 15 miles per hour faster, you would have traveled the same distance in an hour less. How fast did you drive?

Step-by-Step Solution:

Let's denote the speed you drove as x miles per hour.

The time taken to drive 180 miles at this speed can be expressed as:

Time Distance / Speed 180 / x

Furthermore, if you had driven 15 miles per hour faster, your speed would be x 15 miles per hour. The time taken to drive the same distance at this increased speed would be:

Time 180 / (x 15)

According to the problem, driving 15 miles per hour faster would result in traveling the same distance in one hour less. We can set up the equation:

180 / x - 180 / (x 15) 1

Multiplying Through:

Multiply both sides by x(x 15) to eliminate the denominators:

180(x 15) - 18 x(x 15)

Simplifying the Equation:

18 2700 - 18 x^2 15x

This simplifies to:

2700 x^2 15x

Rearranging gives us a standard quadratic equation:

x^2 - 15x - 2700 0

Solving the Quadratic Equation:

We can use the quadratic formula to find x:

x (-b ± √(b^2 - 4ac)) / (2a)

where a 1, b -15, and c -2700.

x (-(-15) ± √((-15)^2 - 4(1)(-2700))) / (2(1))

Calculating the discriminant:

(-15)^2 225

-4(1)(-2700) 10800

b^2 - 4ac 225 10800 11025

Now substitute back into the formula:

x (15 ± √11025) / 2

Calculating √11025:

√11025 105

x (15 ± 105) / 2

Calculating the two possible solutions:

x (15 105) / 2 45 (valid solution)

x (15 - 105) / 2 -60 (not valid)

Therefore, the speed you drove is:

boxed{45 mph}

Real-World Confirmation:

To verify the solution, we can check the consistency:

45 mph for 180 miles takes 4 hours.

60 mph for the same distance takes 3 hours, which is indeed one hour less.

Thus, the speed you drove is indeed 45 mph.

As you can see, algebraic equations provide a powerful tool for solving real-life problems involving speed, distance, and time.