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Answer :
Final answer:
To find how many cycles an airplane can complete in an hour with a sinusoidal altitude path, we convert its horizontal speed from miles per hour to feet, find the length of one cycle in feet from the cosine function's argument, and then divide the total distance traveled by the cycle length. The airplane completes approximately 16.8 cycles in one hour.
Explanation:
To determine how many cycles the airplane can complete in an hour with a path modeled by the function h(x) = 10000cos(0.0001x + 20000), we need to understand the periodic nature of the cosine function and convert the airplane's forward speed into the same units used in the function.
The cosine function has a period of 2π, which means it repeats every 2π radians. Since x represents the forward distance in feet, and we are given the speed in miles per hour, we first need to convert miles to feet (1 mile = 5280 feet). At 200 miles per hour, the airplane travels 200 x 5280 feet per hour. The argument of the cosine function (0.0001x) completes one cycle when x increases by 2π / 0.0001 feet. Therefore, the number of cycles in one hour is the distance traveled in feet divided by the length of one cycle in feet.
Calculating this, we have:
- Total distance traveled in an hour in feet: 200 miles/hour x 5280 feet/mile = 1,056,000 feet.
- Length of one cycle in feet: 2π / 0.0001 ≈ 62,831.85 feet.
- Number of cycles completed in an hour: 1,056,000 feet / 62,831.85 feet/cycle ≈ 16.81 cycles.
Therefore, the airplane completes approximately 16.8 cycles in one hour.
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