Problem 33 - Algebra 2

Question

Motion You are sitting on a pier watching the waves when you notice a bottle in the water. The bottle bobs so that it is between 2.5 \(\mathrm{ft}\) and 4.5 \(\mathrm{ft}\) below the pier. You know you can reach 3 \(\mathrm{ft}\) below the pier. Suppose the bottle reaches its highest point every 5 \(\mathrm{s} .\) a. Sketch a graph of the bottle's distance below the pier for 15 s. Assume that at \(t=0,\) the bottle is closest to the pier. b. Find the period and the amplitude of the function. c. Estimatien Use your graph to estimate the length of time the bottle is within reach during each cycle.

Step-by-Step Solution

Answer

The period of the function is 5 seconds, and the amplitude is 1 ft. Use the graph to estimate the time bottle is in reach during each cycle.

Step 1: Analyzing Given Information

The problem mentions that the distance of the bottle below the pier varies between 2.5 and 4.5 ft. The bottle reaches its highest point every 5 seconds which is when it's closest to the pier. This suggests that the motion of the bottle is periodic.

Step 2: Sketching the Graph

The graph will start at the highest point (closest to pier) because the starting point \(t=0\) is when the bottle is closest. It will rise and fall between 2.5 and 4.5 ft indicating that it moves 1 ft below and above the middle point which is 3.5 ft. The complete cycle repeats every 5 seconds. Sketching this will help visualize the motion of the bottle over the time.

Step 3: Determining the Period and the Amplitude

The problem states the bottle reaches its highest point every 5 seconds, meaning one complete cycle of up and down takes 5 seconds. Therefore, the period of this function is 5 seconds. The amplitude is the maximum distance it moves from the rest position (middle point), which is 1 ft (the distance between 3.5 ft middle point and 2.5 or 4.5 ft).

Step 4: Estimating Time the Bottle is Within Reach

The bottle is in reach when it is less than or equals to 3 ft below the pier. So, by looking at the graph, one can estimate the time for which the bottle is within 3 ft below the pier during each cycle period of 5 seconds.

Key Concepts

Amplitude

When discussing periodic motion, amplitude is a crucial concept. It is the maximum extent of oscillation measured from the equilibrium position. In the context of the bottle bobbing in the water, the amplitude represents how far the bottle moves above or below the middle point of its trajectory.

In this exercise, we know the bottle moves between 2.5 ft and 4.5 ft below the pier. The middle point, or equilibrium position, is precisely halfway between these two values, at 3.5 ft. Thus, the amplitude is the distance from this equilibrium point to either extreme position.

• Amplitude = Difference from Middle Point to Extreme Point.
• In this case, the amplitude is 1 ft, as the bottle reaches up to 1 ft above and below the middle point (from 3.5 ft to 2.5 ft and 4.5 ft).

Amplitude is a handy measure, as it helps gauge the energy or intensity of oscillations in a periodic motion. Knowing the amplitude allows us to understand just how much 'room' the moving object needs to complete its cycles.

Period

The period of a periodic motion is the time taken for a complete cycle to pass a given point. This is a fundamental concept in wave-like movements, where establishing the duration of one complete cycle provides a basis for analyzing the motion.

In our bottle scenario, the period is straightforward as it’s given: the bottle reaches its highest point every 5 seconds. This means that every 5 seconds, the motion resets, and the entire cycle of moving from the highest point to the lowest point and back begins anew.

• Period = Time taken for a complete cycle.
• Here, one cycle of the bottle's movement takes exactly 5 seconds.

Understanding the period is essential because it helps in predicting future positions and movements, making it easier to sketch graphs and plan around the periodic motion's behavior.

Graph Sketching

Graph sketching can be an insightful way to depict periodic motions, especially when trying to visualize changes over time.

To sketch the graph for the bottle bobbing in water:

• Start with the y-axis representing the distance below the pier and the x-axis representing time.
• At time, \( t=0 \), the bottle is closest to the pier at 2.5 ft below.
• As the time progresses to \( t=2.5 \) seconds, the bottle reaches the farthest point from the pier at 4.5 ft below.
• By \( t=5 \) seconds, the bottle returns to 2.5 ft, completing one cycle.
• This cycle repeats every 5 seconds, maintaining a sinusoidal pattern, due to symmetry and periodic nature.

Visualizing this pattern helps estimate moments when the bottle remains within reach of 3 ft. It shows that through each cycle, there are portions less than 3 ft, reflecting the duration for which you can possibly reach the bottle. Graphically representing the motion assists in interpreting and utilizing the periodic nature effectively in practical scenarios.