Math / Trigonometry

Questionweight is attached to a spring suspended from a beam. At time $t=0$ , it is pulled down to a point 7 cm above the ground and leased. After that, it bounces up and down between its minimum height of 7 cm and a maximum height of 25 cm , and its height $(t)$ is a sinusoidal function of time $t$ . It first reaches a maximum height 0.8 seconds after starting. (a) Follow the procedure outlined in this section to sketch a rough graph of $h(t)$ . Draw at least two complete cycles of the oscillation, indicating where the maxima and minima occur. (b) What are the mean, amplitude, phase shift and period for this function? (Use a phase shift with an absolute value less than the period.) mean $\square$ amplitude $\square$ phase shift $\square$ period $\square$ (c) Give four different possible values for the phase shift. (Enter your answers as a comma-separated list.) $\square$ (d) Write down a formula for the function $h(t)$ in standard sinusoidal form; i.e. as in the equation shown below. $\begin{array}{l} h(t)=A \sin \left(\frac{2 \pi}{B}(t-C)\right)+D \\ h(t)=\square \end{array}$ (e) What is the height of the weight after 2.4 seconds? $\square$ cm (f) During the first 10 seconds, how many times will the weight be exactly 19 cm above the floor? (Note: This problem does not require inverse trigonometry.) $\square$

Studdy Solution

STEP 1

1. The height $h(t)$ is a sinusoidal function of time $t$ .
2. The minimum height is 7 cm, and the maximum height is 25 cm.
3. The first maximum height is reached at $t = 0.8$ seconds.

STEP 2

1. Sketch the graph of $h(t)$ .
2. Determine the mean, amplitude, phase shift, and period of $h(t)$ .
3. Calculate four possible phase shifts.
4. Write the formula for $h(t)$ in standard sinusoidal form.
5. Calculate the height after 2.4 seconds.
6. Determine how many times the weight is exactly 19 cm above the floor in the first 10 seconds.

STEP 3

To sketch the graph, note that the function is sinusoidal with a minimum at 7 cm and a maximum at 25 cm. The mean height is the average of these two values.
$\text{Mean} = \frac{7 + 25}{2} = 16 \, \text{cm}$
The amplitude is the difference between the mean and the maximum (or minimum).
$\text{Amplitude} = 25 - 16 = 9 \, \text{cm}$
The function starts at a minimum at $t = 0$ , reaches a maximum at $t = 0.8$ , and completes a full cycle when it returns to a minimum. Thus, the period is twice the time to reach the maximum.
$\text{Period} = 2 \times 0.8 = 1.6 \, \text{seconds}$

STEP 4

The mean is 16 cm, the amplitude is 9 cm, and the period is 1.6 seconds. The phase shift is determined by the starting point of the cycle.
Since the function starts at a minimum, the phase shift is $-\frac{1.6}{4} = -0.4 \, \text{seconds}$ .

STEP 5

Four possible phase shifts can be calculated by adding or subtracting the period:
$-0.4, 1.2, 1.6 - 0.4 = 1.2, 1.6 + 0.8 = 2.4$
Possible phase shifts: $-0.4, 1.2, 2.0, 2.4$

STEP 6

The standard sinusoidal form is:
$h(t) = A \sin \left(\frac{2\pi}{B}(t - C)\right) + D$
Substituting the known values:
$h(t) = 9 \sin \left(\frac{2\pi}{1.6}(t + 0.4)\right) + 16$

STEP 7

To find the height at $t = 2.4$ seconds, substitute into the formula:
$h(2.4) = 9 \sin \left(\frac{2\pi}{1.6}(2.4 + 0.4)\right) + 16$
Calculate the sine value and solve for $h(2.4)$ .

STEP 8

After calculation, the height at $t = 2.4$ seconds is approximately 16 cm, as it is at the mean position.

STEP 9

To determine how many times the weight is exactly 19 cm above the floor in the first 10 seconds, solve the equation:
$19 = 9 \sin \left(\frac{2\pi}{1.6}(t + 0.4)\right) + 16$
This simplifies to:
$\sin \left(\frac{2\pi}{1.6}(t + 0.4)\right) = \frac{1}{3}$
Calculate the number of solutions within the interval $0 \leq t \leq 10$ .
The number of times the weight is exactly 19 cm above the floor is 12.

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Studdy Solution

STEP 1

1. The height $h(t)$ is a sinusoidal function of time $t$ .
2. The minimum height is 7 cm, and the maximum height is 25 cm.
3. The first maximum height is reached at $t = 0.8$ seconds.

STEP 2

STEP 3

STEP 4

The mean is 16 cm, the amplitude is 9 cm, and the period is 1.6 seconds. The phase shift is determined by the starting point of the cycle.
Since the function starts at a minimum, the phase shift is $-\frac{1.6}{4} = -0.4 \, \text{seconds}$ .

STEP 5

Four possible phase shifts can be calculated by adding or subtracting the period:
$-0.4, 1.2, 1.6 - 0.4 = 1.2, 1.6 + 0.8 = 2.4$
Possible phase shifts: $-0.4, 1.2, 2.0, 2.4$

STEP 6

The standard sinusoidal form is:
$h(t) = A \sin \left(\frac{2\pi}{B}(t - C)\right) + D$
Substituting the known values:
$h(t) = 9 \sin \left(\frac{2\pi}{1.6}(t + 0.4)\right) + 16$

STEP 7

To find the height at $t = 2.4$ seconds, substitute into the formula:
$h(2.4) = 9 \sin \left(\frac{2\pi}{1.6}(2.4 + 0.4)\right) + 16$
Calculate the sine value and solve for $h(2.4)$ .

STEP 8

After calculation, the height at $t = 2.4$ seconds is approximately 16 cm, as it is at the mean position.

STEP 9

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Studdy Solution

STEP 1

1. The height h(t) h(t) h(t) is a sinusoidal function of time t t t.2. The minimum height is 7 cm, and the maximum height is 25 cm.3. The first maximum height is reached at t=0.8 t = 0.8 t=0.8 seconds.

STEP 2

STEP 3

STEP 4

The mean is 16 cm, the amplitude is 9 cm, and the period is 1.6 seconds. The phase shift is determined by the starting point of the cycle.Since the function starts at a minimum, the phase shift is −1.64=−0.4 seconds -\frac{1.6}{4} = -0.4 \, \text{seconds} −41.6​=−0.4seconds.

STEP 5

Four possible phase shifts can be calculated by adding or subtracting the period:−0.4,1.2,1.6−0.4=1.2,1.6+0.8=2.4 -0.4, 1.2, 1.6 - 0.4 = 1.2, 1.6 + 0.8 = 2.4 −0.4,1.2,1.6−0.4=1.2,1.6+0.8=2.4Possible phase shifts: −0.4,1.2,2.0,2.4-0.4, 1.2, 2.0, 2.4−0.4,1.2,2.0,2.4

STEP 6

The standard sinusoidal form is:h(t)=Asin⁡(2πB(t−C))+D h(t) = A \sin \left(\frac{2\pi}{B}(t - C)\right) + D h(t)=Asin(B2π​(t−C))+DSubstituting the known values:h(t)=9sin⁡(2π1.6(t+0.4))+16 h(t) = 9 \sin \left(\frac{2\pi}{1.6}(t + 0.4)\right) + 16 h(t)=9sin(1.62π​(t+0.4))+16

STEP 7

To find the height at t=2.4 t = 2.4 t=2.4 seconds, substitute into the formula:h(2.4)=9sin⁡(2π1.6(2.4+0.4))+16 h(2.4) = 9 \sin \left(\frac{2\pi}{1.6}(2.4 + 0.4)\right) + 16 h(2.4)=9sin(1.62π​(2.4+0.4))+16Calculate the sine value and solve for h(2.4) h(2.4) h(2.4).

STEP 8

After calculation, the height at t=2.4 t = 2.4 t=2.4 seconds is approximately 16 cm, as it is at the mean position.

STEP 9

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1. The height $h(t)$ is a sinusoidal function of time $t$ .
2. The minimum height is 7 cm, and the maximum height is 25 cm.
3. The first maximum height is reached at $t = 0.8$ seconds.

The mean is 16 cm, the amplitude is 9 cm, and the period is 1.6 seconds. The phase shift is determined by the starting point of the cycle.
Since the function starts at a minimum, the phase shift is $-\frac{1.6}{4} = -0.4 \, \text{seconds}$ .

Four possible phase shifts can be calculated by adding or subtracting the period:
$-0.4, 1.2, 1.6 - 0.4 = 1.2, 1.6 + 0.8 = 2.4$
Possible phase shifts: $-0.4, 1.2, 2.0, 2.4$

The standard sinusoidal form is:
$h(t) = A \sin \left(\frac{2\pi}{B}(t - C)\right) + D$
Substituting the known values:
$h(t) = 9 \sin \left(\frac{2\pi}{1.6}(t + 0.4)\right) + 16$

To find the height at $t = 2.4$ seconds, substitute into the formula:
$h(2.4) = 9 \sin \left(\frac{2\pi}{1.6}(2.4 + 0.4)\right) + 16$
Calculate the sine value and solve for $h(2.4)$ .

After calculation, the height at $t = 2.4$ seconds is approximately 16 cm, as it is at the mean position.