Problem 85 - Algebra A Combined Function

Question

A hyena spots a giraffe 0.5 mile away and begins running toward it. The giraffe starts running away from the hyena just as the hyena begins running toward it. A hyena can run at a speed of \(40 \mathrm{mph}\) and a giraffe can run at \(32 \mathrm{mph}\). How long will it take the hyena to overtake the giraffe? (Source: The World Almanac and Book of Facts)

Step-by-Step Solution

Answer

It takes the hyena 3.75 minutes to overtake the giraffe.

Step 1: Understand the Relative Speed

To solve this problem, we first need to understand the concept of relative speed when two objects are moving in opposite directions.The relative speed between the hyena and the giraffe is the difference in their speeds, because they are moving in opposite directions. The hyena runs at a speed of 40 mph, and the giraffe runs at a speed of 32 mph. Therefore, the relative speed between the hyena and the giraffe is:\[\text{Relative Speed} = 40 \text{ mph} - 32 \text{ mph} = 8 \text{ mph}\]

Step 2: Calculate Time Using Distance and Relative Speed

With the relative speed calculated, we can now find out how much time it will take for the hyena to catch up to the giraffe.The distance between them initially is 0.5 miles. Using the formula: \[\text{Time} = \frac{\text{Distance}}{\text{Relative Speed}}\]Substitute the known values:\[\text{Time} = \frac{0.5 \text{ miles}}{8 \text{ mph}} = 0.0625 \text{ hours}\]

Step 3: Convert Time to Minutes

To make the answer more intuitive, we convert the time from hours to minutes, given that 1 hour equals 60 minutes.\[\text{Time in Minutes} = 0.0625 \text{ hours} \times 60 \text{ minutes/hour} = 3.75 \text{ minutes}\]

Key Concepts

Distance Formula

The distance formula is a simple but fundamental concept in physics and mathematics. It's used to calculate the distance traveled by an object when you know its speed and the time it took to travel that distance. The basic formula is:\[\text{Distance} = \text{Speed} \times \text{Time}\]In the context of the exercise with the hyena and giraffe, although the precise calculation to determine when the hyena overtakes the giraffe does not directly use the distance formula, understanding it is important. Knowing the initial 0.5 miles between the animals, and the relative speed, aids in determining the time for the hyena to reach the giraffe. The calculation primarily involves working backwards using speed and time, showing how versatile the distance formula can be in different scenarios beyond just simple distance calculation.

Speed and Velocity

Speed and velocity are crucial concepts in understanding motions like those of the hyena and the giraffe. Speed is a scalar quantity, meaning it only has magnitude and no direction. It tells us how fast an object is moving.

• In this problem, the hyena runs at \(40 \text{ mph}\) and the giraffe at \(32 \text{ mph}\).
• These speeds give us a sense of how quickly each animal covers a distance.

Velocity, on the other hand, is a vector quantity, which means it includes both speed and direction. While the problem deals mostly with speed because direction is straightforward and implicit (running towards or away), understanding velocity can be significant in more complex scenarios. Here, relative speed is used to understand the effective speed at which the hyena approaches the giraffe. By calculating \(40 \text{ mph} - 32 \text{ mph} = 8 \text{ mph}\), we find that the hyena effectively "closes in" on the giraffe at \(8 \text{ mph}\).

Unit Conversion

Unit conversion is often necessary when solving problems to make the final answer more meaningful or to comply with the question's requirements. Typically, you'll convert units like distance, time, or speed to gain better insights. In this exercise, converting time from hours to minutes helps provide a more intuitive answer.

• The initial calculation gives us \(0.0625\) hours, but this doesn't easily convey how long it really is to most people.
• By converting \(0.0625\) hours into minutes: \(0.0625 \text{ hours} \times 60 \text{ minutes/hour} = 3.75 \text{ minutes}\), the time more conveniently signifies how quickly the hyena reaches the giraffe.

Effective unit conversion ensures that anyone reading the solution understands the result, regardless of their familiarity with the problem's initial units. This skill is particularly useful in day-to-day problem solving and when communicating findings to others.