At a high-altitude observatory, Emek and Tara prepared a rotating telescope to track a distant star. Its viewing angle was modeled by A(t) = 30 sin(πt/4) + 45, where A was the angle in degrees and t was the number of hours after 8:00 p.m. Tara identified the amplitude as 30 degrees, so the telescope moved between 15 and 75 degrees. Emek determined that the period was 8 hours, meaning the same pattern repeated every 8 hours. The star would become visible when the telescope first reached its maximum angle of 75 degrees. Since the sine function reaches its maximum when its input is π/2, they solved πt/4 = π/2 and obtained t = 2. They checked the graph and confirmed that the angle started at 45 degrees, rose to 75 degrees after 2 hours, and returned to 45 degrees after 4 hours. Knowing the required orientation would occur at 10:00 p.m., Emek and Tara prepared the instruments and began observing the star at exactly the right moment.

Challenges

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Challenge 1 of 5

Challenges

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Challenge 1 of 4
Hint
In a model of the form a sin(bt) + d, the amplitude is the absolute value of a and the midline is d.
For the telescope model A(t) = 30 sin(πt/4) + 45, which statement correctly describes its amplitude, midline, and range?
Amplitude 45 degrees, midline 30 degrees, range from -15 to 75 degreesAmplitude 30 degrees, midline 45 degrees, range from 15 to 75 degreesAmplitude 30 degrees, midline 75 degrees, range from 45 to 105 degreesAmplitude 45 degrees, midline 45 degrees, range from 0 to 90 degrees
Correct! The amplitude is 30 degrees and the midline is 45 degrees, giving a range from 15 to 75 degrees.
Not quite. Identify the amplitude and vertical shift, then use them to find the minimum and maximum angles.
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Hint
For a sine cycle, the minimum occurs half a period after the maximum. The model has an 8-hour period.
The telescope follows A(t) = 30 sin(πt/4) + 45. After reaching its first maximum at t = 2 hours, how many additional hours will pass before it reaches its minimum angle for the first time?
Correct! Half of the 8-hour period is 4 hours, so the minimum occurs 4 hours after the maximum.
Check the period of the model and determine how much of a cycle separates a maximum from the next minimum.
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Hint
Start with the first maximum at t = 2 and add one full period each time.
The telescope first reaches its maximum angle at t = 2 hours, and its motion repeats every 8 hours. If the observatory operates for 18 hours after 8:00 p.m., at which values of t will the telescope reach its maximum angle?
t = 2, 6, 10, 14, 18t = 2, 8, 14t = 2, 10, 18t = 2, 4, 6, 8
Correct! Adding the 8-hour period gives maximum angles at t = 2, 10, and 18 hours.
Not quite. Consecutive maximum angles are separated by one complete 8-hour period.
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Dönen Teleskobu Takip Etmek
TR
ARC_G12_GM_02_STORY_01
Dönen Teleskobu Takip Etmek
Tracking the Rotating Telescope
EN
ARC_G12_GM_02_STORY_01
Tracking the Rotating Telescope