Sketch a height versus time graph of the sinusoidal function that models each situation. draw at lease three cycles. assume that the first point plotted on each graph is at the lowest point:
a girl lying on a an air mattress in a wave pool that is 3 m deep, with waves 0.5 m in height that occur at 7 s
i was just wondering if i would start off at -0.5 m, and then at 3.5 s, i would plot at point at 0.5 m, and then at 7 s, i would plot at point at -0.5 again? in that case would 7 be considered the period of the function?
They mean 7 s is the period I believe.
Remember the depth of the pool, 3 meters.
I assume by wave height they mean the amplitude and not the height from bottom to peak which would be 1 here
First point is at 3-.5 = 2.5 m
t---- h
0000 2.5
1.75 3
3.50 3.5
5.25 3
7.00 2.5
8.75 3
10.5 3.5 etc
thank you
but, i would assume the height of the wave would be -0.5 to 0.5 because wouldn't be calculate it in terms of the water's surface?
Yes, you are right. The height versus time graph for the given situation can be modeled as a sinusoidal function. Since the wave pool has waves of 0.5 m in height that occur at a period of 7 s, we can assume a sinusoidal function of the form:
$h(t) = A \sin(\omega t + \phi) + C$
Where:
- A is the amplitude, which is equal to half the difference between the maximum and minimum values of the function. In this case, the amplitude is 0.5 m.
- ω is the angular frequency, which is equal to 2π divided by the period (T) of the function. In this case, T = 7 s, so ω = 2π/7.
- φ is the phase shift, which represents a horizontal shift in the graph. Since we start at the lowest point, φ = 0.
- C is the vertical shift, which represents a shift up or down in the graph. In this case, since the girl is lying on an air mattress 3 m deep, C = 3.
Based on this, the specific function for this situation would be:
$h(t) = 0.5\sin\left(\frac{2\pi}{7}t\right) + 3$
To sketch the graph, we can plot points at intervals of 7 seconds and repeat the pattern for at least three cycles. The first point will be at the lowest point, which is -0.5 m.
Here's a step-by-step sketch of the graph for three cycles:
1. t = 0 s --> h(0) = 0.5sin(0) + 3 = 2.5 m
2. t = 7 s --> h(7) = 0.5sin(2π) + 3 = 2.5 m
3. t = 14 s --> h(14) = 0.5sin(4π) + 3 = 2.5 m
4. t = 21 s --> h(21) = 0.5sin(6π) + 3 = 2.5 m
5. t = 28 s --> h(28) = 0.5sin(8π) + 3 = 2.5 m
6. t = 35 s --> h(35) = 0.5sin(10π) + 3 = 2.5 m
7. t = 42 s --> h(42) = 0.5sin(12π) + 3 = 2.5 m
Repeat this pattern for three cycles, and you will observe a sinusoidal curve with an amplitude of 0.5 m, oscillating around a vertical shift of 3 m. The period of the function is indeed 7 s, as it takes 7 seconds for one complete cycle.
To sketch the height versus time graph for the given situation, we need to consider the properties of the sinusoidal function.
First, let's establish the equation of the sinusoidal function based on the given information. The general form of a sinusoidal function is:
f(t) = A * sin(B(t - C)) + D
where:
A is the amplitude (0.5 m in this case),
B is the period (time taken for one complete cycle, which is 7 s in this case),
C is the horizontal phase shift (time taken to start the first cycle, which is 0 s in this case),
D is the vertical phase shift (the lowest point, initially at -0.5 m in this case).
With this information, the equation for the situation becomes:
f(t) = 0.5 * sin((2π/7)(t - 0)) - 0.5
Now, let's plot the graph with at least three cycles:
1. Start the graph at the lowest point (-0.5 m), which corresponds to time t = 0 s (as given).
2. The first complete cycle ends at t = 7 s, which takes the wave pool to its initial state. Thus, we need three cycles, which means ending at t = 21 s.
3. Calculate the height (f(t)) at various points within each cycle.
- At t = 0 s, f(0) = 0.5 * sin((2π/7)(0 - 0)) - 0.5 = -0.5 m (lowest point)
- At t = 3.5 s, f(3.5) = 0.5 * sin((2π/7)(3.5 - 0)) - 0.5 = 0.5 m (middle point)
- At t = 7 s, f(7) = 0.5 * sin((2π/7)(7 - 0)) - 0.5 = -0.5 m (lowest point)
- Similar calculations can be done for other points within each cycle.
4. Repeat step 3 for the next two cycles until t = 21 s, maintaining the pattern.
5. Plot the calculated points on the graph and connect them to form the curve.
Note: The period of the function is indeed 7 s because it represents the time taken for one complete cycle.