Math / Algebra

Question8. A contestant on a game show spins a wheel that is located ona plane perpendicular to the floor. He grabs the only red peg ons the circumference of the wheel, which is 1.5 m above the floor, and pushes it downward. The red peg reaches a minimum height of 0.25 m above the floor and a maximum height of 2.75 m above the floor. Sketch two cycles of the graph that represents the height of the red peg above the floor, as a function of the total distance it moved. Then determine the equation of the sine function that describes the graph.

Studdy Solution

STEP 1

1. The wheel spins in a vertical plane, and the height of the peg varies sinusoidally.
2. The minimum height of the peg is 0.25 m, and the maximum height is 2.75 m.
3. The initial height of the peg is 1.5 m.
4. We need to sketch the graph for two cycles and find the sine function equation.

STEP 2

1. Determine the amplitude, midline, and period of the sine function.
2. Sketch the graph of the height of the red peg for two cycles.
3. Write the equation of the sine function.

STEP 3

Calculate the amplitude of the sine function. The amplitude is half the difference between the maximum and minimum heights.
$\text{Amplitude} = \frac{2.75 - 0.25}{2} = 1.25$

STEP 4

Determine the midline of the sine function. The midline is the average of the maximum and minimum heights.
$\text{Midline} = \frac{2.75 + 0.25}{2} = 1.5$

STEP 5

Determine the period of the sine function. Since we are sketching two cycles, assume one full rotation corresponds to one cycle. The period is the distance for one complete cycle.
Assume the wheel completes one full rotation in a certain distance, say $D$ .

STEP 6

Sketch the graph for two cycles: - Start at the initial height of 1.5 m. - The graph will oscillate between 0.25 m and 2.75 m. - The midline is at 1.5 m. - Complete two full cycles, showing the sinusoidal pattern.

STEP 7

Write the equation of the sine function. The general form is:
$h(x) = A \sin(Bx + C) + D$
Where: - $A = 1.25$ (amplitude) - $D = 1.5$ (midline) - $B$ is determined by the period $\frac{2\pi}{B} = D$ - $C$ is the phase shift, which can be adjusted based on the starting point.
Assuming no phase shift and $D = 2\pi$ for simplicity:
$h(x) = 1.25 \sin\left(\frac{2\pi}{D} x\right) + 1.5$
The equation of the sine function that describes the graph is:
$h(x) = 1.25 \sin\left(\frac{2\pi}{D} x\right) + 1.5$

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Studdy Solution

STEP 1

1. The wheel spins in a vertical plane, and the height of the peg varies sinusoidally.
2. The minimum height of the peg is 0.25 m, and the maximum height is 2.75 m.
3. The initial height of the peg is 1.5 m.
4. We need to sketch the graph for two cycles and find the sine function equation.

STEP 2

1. Determine the amplitude, midline, and period of the sine function.
2. Sketch the graph of the height of the red peg for two cycles.
3. Write the equation of the sine function.

STEP 3

Calculate the amplitude of the sine function. The amplitude is half the difference between the maximum and minimum heights.
$\text{Amplitude} = \frac{2.75 - 0.25}{2} = 1.25$

STEP 4

Determine the midline of the sine function. The midline is the average of the maximum and minimum heights.
$\text{Midline} = \frac{2.75 + 0.25}{2} = 1.5$

STEP 5

Determine the period of the sine function. Since we are sketching two cycles, assume one full rotation corresponds to one cycle. The period is the distance for one complete cycle.
Assume the wheel completes one full rotation in a certain distance, say $D$ .

STEP 6

Sketch the graph for two cycles: - Start at the initial height of 1.5 m. - The graph will oscillate between 0.25 m and 2.75 m. - The midline is at 1.5 m. - Complete two full cycles, showing the sinusoidal pattern.

STEP 7

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Studdy Solution

STEP 1

1. The wheel spins in a vertical plane, and the height of the peg varies sinusoidally.2. The minimum height of the peg is 0.25 m, and the maximum height is 2.75 m.3. The initial height of the peg is 1.5 m.4. We need to sketch the graph for two cycles and find the sine function equation.

STEP 2

1. Determine the amplitude, midline, and period of the sine function.2. Sketch the graph of the height of the red peg for two cycles.3. Write the equation of the sine function.

STEP 3

Calculate the amplitude of the sine function. The amplitude is half the difference between the maximum and minimum heights.Amplitude=2.75−0.252=1.25 \text{Amplitude} = \frac{2.75 - 0.25}{2} = 1.25 Amplitude=22.75−0.25​=1.25

STEP 4

Determine the midline of the sine function. The midline is the average of the maximum and minimum heights.Midline=2.75+0.252=1.5 \text{Midline} = \frac{2.75 + 0.25}{2} = 1.5 Midline=22.75+0.25​=1.5

STEP 5

Determine the period of the sine function. Since we are sketching two cycles, assume one full rotation corresponds to one cycle. The period is the distance for one complete cycle.Assume the wheel completes one full rotation in a certain distance, say D D D.

STEP 6

Sketch the graph for two cycles: - Start at the initial height of 1.5 m. - The graph will oscillate between 0.25 m and 2.75 m. - The midline is at 1.5 m. - Complete two full cycles, showing the sinusoidal pattern.

STEP 7

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Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. The wheel spins in a vertical plane, and the height of the peg varies sinusoidally.
2. The minimum height of the peg is 0.25 m, and the maximum height is 2.75 m.
3. The initial height of the peg is 1.5 m.
4. We need to sketch the graph for two cycles and find the sine function equation.

1. Determine the amplitude, midline, and period of the sine function.
2. Sketch the graph of the height of the red peg for two cycles.
3. Write the equation of the sine function.

Calculate the amplitude of the sine function. The amplitude is half the difference between the maximum and minimum heights.
$\text{Amplitude} = \frac{2.75 - 0.25}{2} = 1.25$

Determine the midline of the sine function. The midline is the average of the maximum and minimum heights.
$\text{Midline} = \frac{2.75 + 0.25}{2} = 1.5$

Determine the period of the sine function. Since we are sketching two cycles, assume one full rotation corresponds to one cycle. The period is the distance for one complete cycle.
Assume the wheel completes one full rotation in a certain distance, say $D$ .