Question

Will Rogers spun a lasso in a vertical circle. The diameter of the loop was 6 ft, and the loop spun 50 times each minutes. If the lowest point on the rope was 6 inches above the ground, write an equation to describe the height of this point above the ground after t seconds.

Please write nicely.

Answer 1

This is a question based on modelling with trignometric functions.

here the trignometric functions used are sine and the cosine functions.

An equation in sine/ cosine is generally of the form :

y=acos(b(x-c))+d or y=asin(b(x-c))+d

where |a| is the amplitude; (2π/b) is the period ; C is horizontal shift and d is vertical shift.

The rope spun 50 times per minute I.e. it completed 50 revolutions in 60 seconds

1 revolution is completed in

Hence, the period is   

Here, there is no horizontal shift , thus,

  

• the diameter of circle =6 feet

1 ft= 12 inches 6 inches=0.5 ft

Hence, the lowest point on rope is m which is at a height of 6 inches(=0.5 ft)

The highest point M is at a height of (0.5+ 6) ft = 6.5 ft

(Because we add the height of lowest point and the diameter of circle.)

Now , coming to finding values of a and d

The amplitude

Also, the minimum height is attained at t=0 , this means that the function is a COSINE function with negative amplitude I.e. a<0

Now, the vertical shift is

Thus, the height of lowest point after time t seconds is h(t)

in feets  

any doubts bug you , feel free to comment your query.

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