Question

Andres boards a Ferris wheel at the 3-o'clock position and rides the Ferris wheel for multiple revolutions. The Ferris wheel rotates at a constant angular speed of 4.7 radians per minute and has a radius of 30 feet. Let t represent the number of seconds since the Ferris wheel started rotating. a. Write an expression (in terms of t) to represent the varying number of radians 8 Ryan has swept out since the ride started. Preview s height (in feet) above Preview c. Write an expression (in terms of t) to represent Ryan's height (in feet) above the ground.

Answer 1

Answer:

a) r(t) = 0.783t radians

b) h(t) = 30cos(0.783*t) feet

Step-by-step explanation:

The situation is depicted in the picture attached

(see picture)

Since the angular speed is constant, to find an expression for the angle r(t) in radians we just cross-multiply using the fact that 1 min = 60 seconds

4.7 radians __________ 60 seconds

r(t) radians ____________  t seconds

$\large \displaystyle\frac{4.7}{r(t)}=\displaystyle\frac{60}{t}\Rightarrow r(t)=(4.7/60)t \Rightarrow\\\\\boxed{r(t)=0.783t\;rad}$

The height h(t) after t seconds is given by

h(t) = 30cos(r(t)) ===>

h(t) = 30cos(0.783*t) feet

Answer 2

Given that Andreas boards the Ferris wheel rotating at 4.7 rad/min at 3-O'clock, we have;

a. r(t) = 7.83 × 10^(-2)•t

c. s = 30•sin((7.83 × 10^(-2))•(t + 40.11)) + 30

How can the rotation and height function be found?

The position Andres boards the Ferris wheel = 3-O'clock

Speed of the Ferris wheel = 4.7 rad/min

Radius of the Ferris wheel = 30 feet

Solution;

a. The speed of the Ferris wheel = 4.7÷60 = 7.83 × 10^(-2)

Number of radians swept out is therefore;

• r(t) = 7.83 × 10^(-2) × t

c. 1 rad in 60÷4.7 = 12.77

2•π rad in 12.77 × 2•π = 80.211

The height of the Ferris wheel is given by the sinusoidal function as follows;

y = A•sin(B•(t + C)) + D

Where;

A = The amplitude = The radius = 30 feet

B = 2•π/T

Therefore;

B = 1/ 12.77 = 7.83 × 10^(-2)

D = 30

At t = 0, y = 30, which gives;

y = 30•sin(B•(0+ C)) + 30

30•sin(B•(0+ C)) = 0

sin(7.83 × 10^(-2)•(C)) = 0

7.83 × 10^(-2)•(C)) = π

C = π ÷ (7.83 × 10^(-2)) = 40.11

The expression to represent Ryan's height is therefore;

• s = 30•sin((7.83 × 10^(-2))•(t + 40.11)) + 30

Learn more about sinusoidal function here:

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