Problem 53 - Elementary Algebra Within Reach

Question

A bubble rises through water at a rate of about \(1.15\) feet per second. How far will the bubble rise in 5 seconds?

Step-by-Step Solution

Answer

The bubble will rise \(5.75\) feet in 5 seconds.

Step 1: Identify given information

The speed at which the bubble rises is \(1.15\) feet per second and the time is \(5\) seconds.

Step 2: Apply the formula for distance

The formula for distance is speed times time or distance = speed * time.

Step 3: Substitute the given values into the formula

Plugging in the given values, we get distance = \(1.15 ft/s * 5 s\)

Step 4: Perform the calculation

By calculating, the distance is \(5.75\) feet

Key Concepts

Understanding Speed

Speed is a measure of how fast an object is moving. It describes the distance you can cover in a certain amount of time. In our bubble example, the speed is given as \(1.15\) feet per second. This means that every second, the bubble travels \(1.15\) feet upwards through the water.

To understand speed, think of it as the rate of movement. When discussing speed, it’s always important to have both a numerical value and a unit of measurement (like feet per second). This tells you not only how much distance is covered, but also how quickly it is being covered.

• The greater the speed, the faster an object travels over a distance.
• If the speed is zero, the object remains stationary.
• Speed is often confused with velocity, but velocity also includes direction, whereas speed does not.

It’s helpful to remember that you can calculate speed by dividing the distance by the time if those measurements are known.

The Role of Time

Time is a crucial factor in calculating distance because it allows us to understand how long an action takes. In this bubble scenario, we know the bubble rises for \(5\) seconds. This gives us a clear duration to apply the distance formula.



Time can be thought of as a constant ticking measure, which helps us quantify the changes in movement or speed over a certain period.

• Longer time periods usually result in more distance traveled if the speed remains constant.
• Time is a scalar quantity - it doesn’t have a direction, unlike vectors such as speed or distance in certain contexts.

When using time in calculations, always make sure it's in the correct unit as compatible with those of speed. Here, since we're using feet per second, the time is also in seconds and thus directly compatible.

Exploring the Distance Formula

The distance formula is a simple equation linking speed, time, and distance: \[\text{Distance} = \text{Speed} \times \text{Time}\]This formula underscores the relationship between how fast something is moving (speed), how long it's been moving (time), and how far it has gone (distance). In the exercise, we used the formula as follows:

• Given speed is \(1.15\) feet/second
• Given time is \(5\) seconds

By substituting these values into the formula, we calculated the distance: \[\text{Distance} = 1.15 \times 5 = 5.75 \text{ feet}\]This shows that after \(5\) seconds, the bubble rises \(5.75\) feet.

The distance formula is valuable in numerous situations, as long as the speed remains constant. Always ensure that all units match (e.g., seconds with seconds, feet with feet) to guarantee accurate calculations. Understanding how to manipulate and apply this formula can help solve real-world problems, just like the bubble rising scenario.