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A Ferris wheel has a 44-foot radius and the center of the Ferris wheel is 50 feet above the ground. The Ferris wheel rotates in the CCW direction at a constant angular speed of 3 radians per minute. Josh boards the Ferris wheel at the 3-o'clock position and rides the Ferris wheel for many rotations. Let tt represent the number of minutes since the ride started.

Write an expression (in terms of t) to represent the number of radians Josh has swept out from the 3-o'clock position since the ride started.
   

How long does it take for Josh to complete one full revolution (rotation)?
________ minutes   

Write an expression (in terms of t) to represent Josh's height above the center of the Ferris wheel (in feet).
   

Write a function f that determine's Josh's height above the ground (in feet) in terms of t.
f(t)=

​

Respuesta :

Answer:

Part A

The expression in terms of 't' representing the number of radians Josh has swept since the ride started is 3·t rad

Part B

The time to complete one revolution is (2·π/3) minutes

Part C.

The function 'f' that determines Josh's height above the ground (in feet) in terms of 't' is, f(t) = 44·sin(3·t + π/2) + 50

Step-by-step explanation:

The radius of the Ferris wheel = 44 foot

The height of the Ferris wheel above the ground = 50 feet

The direction of rotation of the Ferris wheel = CCW

The angular speed of the Ferris wheel, ω = 3 radians per minute

The position from which Josh boards the Ferris wheel = 3-O'Clock

The number of minutes since the ride started started = t

Part A

The expression in terms of 't' to represent the number of radians Josh has swept since the ride started = ω × t = 3 × t rad = 3·t rad

Part B

One complete revolution = 2·π radians

The time to complete one revolution = 2·π/ω = 2·π/(3 rad/minute) = (2·π/3) minutes =  [tex]0.\overline{6}[/tex]·π minutes ≈ 2.0943951 minutes

Part C.

The height, 'h', of the Ferris wheel is given by the following sinusoidal relation

f(t) = A·sin(B·t + C) + D

Where;

A = The amplitude = The radius of the Ferris wheel = 44 ft.

The period, T = (2·π/3) minute

B = 2·π/T = 3

t = The time in minutes

D = The Ferris wheel's height above ground = 50 ft.

C = The horizontal shift = π/2 (Josh boards the Ferris wheel at t = 3-O'Clock which is the midline, a rotation of π/2 from the lowermost end of the Ferris wheel)

Therefore the function that determines Josh's height above the ground (in feet) in terms of 't' is presented as follows;

f(t) = 44·sin(3·t + π/2) + 50

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