Mathematical Modeling Of A Water Ride Finding Time Intervals Below The Starting Point

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Hey guys! Ever wondered how math can actually model something as exciting as a water ride? Today, we're diving deep – both literally and figuratively – into the mathematical representation of a thrilling water ride. We’ll be looking at how the sinusoidal function can map out the ups and downs, twists, and turns of a ride where you might find yourself both high above and deep below the starting point. So, buckle up, and let’s get started on this mathematical adventure!

Understanding the Water Ride Function

Our mathematical journey begins with a function that describes the vertical motion of our water ride: y = 3sin(π/2(x + 3)) - 2. This equation might look a bit intimidating at first, but don't worry, we'll break it down piece by piece. This equation is a sinusoidal function, which is perfect for modeling repetitive, wave-like motion – exactly what you'd expect from a water ride that goes up and down! The 'y' represents the vertical position of the rider relative to a starting point, and 'x' represents the time in seconds. The sine function itself (sin) is the heart of this model, oscillating between -1 and 1. Let's dissect the components of this function to truly grasp their roles.

  • Amplitude (3): The number 3 in front of the sine function is the amplitude. In the context of our water ride, the amplitude determines the maximum vertical displacement from the center line. Since our amplitude is 3, the ride will go up to 3 units above and down to 3 units below the center line of its motion. This gives us a sense of how high and low the ride will swing.
  • Horizontal Stretch (π/2): The term π/2 that multiplies (x + 3) affects the period of the sine function. The period is the length of one complete cycle of the ride – how long it takes to go from the top, down to the bottom, and back to the top again. To find the period, we use the formula 2π divided by the coefficient of x, which in our case is π/2. So, the period is 2π / (π/2) = 4 seconds. This means each full cycle of the ride's motion takes 4 seconds.
  • Horizontal Shift (+3): The '+3' inside the sine function (within the parentheses) represents a horizontal shift, also known as a phase shift. This shifts the entire graph 3 units to the left. In practical terms, this means that the ride’s motion starts as if it had already been running for 3 seconds. It influences where the ride is at time x = 0.
  • Vertical Shift (-2): Finally, the '-2' at the end of the equation is a vertical shift. This moves the entire graph down by 2 units. So, the centerline of the ride's motion is at y = -2, meaning the ride oscillates around a point 2 units below what we might consider the "starting" point.

By understanding each of these components, we can start to visualize the ride's journey – a thrilling sequence of ascents and descents, all perfectly timed and spaced out. This is the power of mathematical modeling! Now, let's move on to finding out when the ride is below the starting point.

Determining When the Ride Dips Below the Starting Point

The million-dollar question now is: when does this thrilling water ride dip below the starting point within the interval of 0 to 7 seconds? Remember, our starting point is y = 0, so we're essentially looking for the times when the function y = 3sin(π/2(x + 3)) - 2 is less than 0. To find these intervals, we need to solve the inequality: 3sin(π/2(x + 3)) - 2 < 0. This might sound complex, but let's break it down step by step. Firstly, we isolate the sine function:

  1. Add 2 to both sides: 3sin(π/2(x + 3)) < 2
  2. Divide by 3: sin(π/2(x + 3)) < 2/3

Now, we're dealing with a standard trigonometric inequality. The next step involves finding the angles for which the sine function is equal to 2/3. This is where the inverse sine function (arcsin) comes in handy. Let's find the principal angle:

  • arcsin(2/3) ≈ 0.7297 radians

However, the sine function is positive in both the first and second quadrants. So, we need to find the corresponding angle in the second quadrant as well. This angle can be found by subtracting the principal angle from π (since π radians = 180 degrees):

  • π - 0.7297 ≈ 2.4119 radians

These two angles represent the points where the ride transitions from being above the starting point to below it, and vice versa, within a single cycle. But remember, our function has a phase shift and a period that we need to account for. Let's incorporate these angles back into our original equation's argument (π/2(x + 3)):

  • π/2(x + 3) = 0.7297 and π/2(x + 3) = 2.4119

Now, we solve for x in each equation to find the times when the ride dips below the starting point:

  1. For π/2(x + 3) = 0.7297:
    • x + 3 = (2/π) * 0.7297
    • x ≈ -2.534
  2. For π/2(x + 3) = 2.4119:
    • x + 3 = (2/π) * 2.4119
    • x ≈ -1.466

These values of x are outside our interval of 0 to 7 seconds, but they give us a starting point. Since the period of our function is 4 seconds, we can add multiples of 4 to these x-values to find solutions within our interval. Let's do that:

  • -2.534 + 4 = 1.466
  • -1.466 + 4 = 2.534
  • -2.534 + 8 = 5.466
  • -1.466 + 8 = 6.534

So, we have four potential times within our interval: 1.466, 2.534, 5.466, and 6.534 seconds. Now, we need to determine the intervals where the ride is below the starting point. The sine function is less than 2/3 between the angles we calculated, so we're looking for the times between our x-values:

  • The ride is below the starting point between approximately 1.466 and 2.534 seconds.
  • The ride is below the starting point again between approximately 5.466 and 6.534 seconds.

Therefore, by carefully dissecting the function and applying trigonometric principles, we've pinpointed the times when the water ride plunges below the starting point, adding another layer of understanding to this thrilling experience.

Identifying the Correct Time Intervals

Alright, let's circle back to our initial question: which time intervals accurately represent when the water ride is below the starting point? We've done the heavy lifting in our calculations, and now it's time to match our findings with the given options. Remember, we determined that the ride is below the starting point between approximately 1.466 and 2.534 seconds, and again between 5.466 and 6.534 seconds. Now, let’s examine the options provided:

  • A. 2.738 and 3.809 seconds: These times don't align with our calculated intervals. We found the ride goes below the starting point earlier than this.
  • B. 2.057 and 4.49 seconds: This option includes a time (2.057 seconds) that falls within our first interval (1.466 and 2.534 seconds). However, 4.49 seconds doesn't fit within our second interval (5.466 and 6.534 seconds).
  • C. 2.49 and 4.057 seconds: This option has 2.49 seconds that does fall within our first interval. Again, 4.057 seconds is nowhere near our second interval.
  • D. 1.809 and 4.65 seconds: This option has 1.809 seconds which nicely sits within our first interval. But 4.65 seconds is off the mark, as it doesn't fall into the second interval we identified.

Considering our derived intervals, the options provided aren't perfectly matching our solutions. However, based on our calculations, the ride initially goes below the starting point around 1.466 seconds and comes back up around 2.534 seconds. None of the options accurately reflect both intervals we found. It seems we might have a slight discrepancy between the provided options and our precise calculations. In a real-world scenario, this could be due to rounding errors or the limitations of the provided choices.

Final Thoughts on Mathematical Modeling of Thrill Rides

So, guys, we've reached the end of our mathematical water ride adventure! We started with a seemingly complex trigonometric function and unraveled it to understand the thrilling journey it represents. We dissected the equation y = 3sin(π/2(x + 3)) - 2, identifying the amplitude, period, phase shift, and vertical shift, and saw how each element contributes to the ride's motion. We then tackled the challenge of finding when the ride dips below the starting point, navigating through trigonometric inequalities and inverse functions. Although the provided options didn't perfectly align with our calculated intervals, we learned the importance of precise calculations and how mathematical models can closely approximate real-world scenarios.

This exercise beautifully illustrates the power of mathematics in modeling and understanding the world around us – even something as exhilarating as a water ride! By breaking down complex problems into manageable steps, we can gain insights and make predictions. So, next time you're on a thrilling ride, remember that there's a whole world of math behind the fun!

Hopefully, this exploration has not only helped clarify the specific problem but also sparked an appreciation for the role of mathematics in everyday experiences. Keep exploring, keep questioning, and keep applying math to unravel the mysteries around you. Until next time, keep the mathematical spirit soaring high!