Problem 23 - Algebra 1

Question

In Exercises \(23-26,\) use this information. A rabbit is 30 feet from its burrow. It can run 25 feet per second. A coyote that can run 50 feet per second spots the rabbit and starts running toward it. Write expressions for the time it will take the rabbit to get to its burrow and for the time it will take the coyote to get to the burrow.

Step-by-Step Solution

Answer

The time for the rabbit to reach its burrow is 1.2 seconds, while the time for the coyote to reach the burrow is 0.6 seconds.

Step 1: Expression for the time it takes the rabbit to reach its burrow

Given that the rabbit's speed is 25 feet per second and the burrow is 30 feet away. Substitute these given values into the formula. Time for rabbit = \(\frac{distance}{speed} = \frac{30}{25} = 1.2\) seconds.

Step 2: Expression for the time it takes the coyote to reach the burrow

Using the same approach for the coyote, whose speed is 50 feet per second, and distance to the burrow is still 30 feet. Substitute these values into the formula. Time for coyote = \(\frac{distance}{speed} = \frac{30}{50} = 0.6\) seconds.

Key Concepts

Speed

Speed is a crucial concept in distance-time calculations. It refers to how fast an object moves per unit of time. When calculating speed, we use the formula: \[\text{Speed} = \frac{\text{Distance}}{\text{Time}} \]This formula helps determine the velocity at which something covers a distance over a specified period. In our exercise, we have two animals: a rabbit and a coyote. - The rabbit runs 25 feet per second. - The coyote moves 50 feet per second.By understanding their speeds, we can figure out how quickly each animal will reach their goal, given a specific distance.

Distance

Distance is simply how far an object travels, irrespective of its starting or ending points. It plays a key role in determining time when you know the speed. In our example, both the rabbit and the coyote aim for the same distance of 30 feet to their burrow. The formula for distance is straightforward when you have speed and time, which is:\[\text{Distance} = \text{Speed} \times \text{Time}\]This helps us visualize how a set distance, like 30 feet, can be covered by different means depending on the speed of the moving objects involved.

Time

Time measures how long it takes to travel a certain distance at a certain speed. Understanding time in our calculations is about using the speed and distance to find out how long a journey will last. The formula we use for figuring out time is:\[\text{Time} = \frac{\text{Distance}}{\text{Speed}}\]For the rabbit:- 30 feet - 25 feet/second speed This results in a time expression of \(1.2\) seconds to get to the burrow.Similarly for the coyote:- 30 feet- 50 feet/second speedThis results in a time expression of \(0.6\) seconds to reach the same destination.These calculations show the importance of speed affecting time.

Expressions

Expressions in mathematics represent equations used to convey relationships between different quantities. In distance-time problems, expressions offer a concise way to show the relationship between distance, speed, and time.In our example:- The expression for the time it takes the rabbit to reach the burrow is \(\frac{30}{25} = 1.2\) seconds.- For the coyote, it is \(\frac{30}{50} = 0.6\) seconds.These expressions are simple yet powerful tools that help to quickly determine the time required for different bodies to cover the same distance. Crafting accurate expressions is a foundational skill in algebra that fosters a deeper understanding of real-world problems.