Question

Emily is on a Ferris wheel and her height is modeled by the equation
27
y = 30.48 cos2pi/4.25 (x - 2) + 34.27 where x = the number of minutes since the
ride began and y = the height in feet of Emily.
a) What is the radius of the Ferris wheel?
b) How long does it take Emily to make one rotation?
The radius is______
feet and it takes Emily_____
minutes for one rotation.

Answer 1

The given function for the height of the Ferris wheel is a sinusoidal equation.

• a) The radius is 30.48 feet and it takes Emily 4.25 minutes for one rotation

Reasons:

The function that gives the height of the Ferris wheel is presented as follows;

$\displaystyle y = 30.48 \cdot cos \frac{2\cdot \pi }{4.25} \cdot \left(x - 2 \right) + 34.27$

Where;

x = The number of minutes since the ride began

y = The height in feet of Emily

a) Required:

The radius of the Ferris wheel

Solution:

Comparing the given equation with the general equation of a Ferris wheel, we have;

$\displaystyle f(t) = \mathbf{A \cdot trig\left(B \cdot \left(t + C\right) \right) + D}$

The amplitude of the Ferris wheel = A = 30.48

The amplitude, A = The largest distance from the center = The radius, r

A = r

Therefore;

The radius of the Ferris wheel, r = A = 30.48 feet

b) The time it takes the Emily to make one rotation is given by the period, T, of the motion of the Ferris wheel as follows;

$\displaystyle T = \mathbf{\frac{2 \cdot \pi}{B}}$

By comparison, we have;

$\displaystyle B = \frac{2 \cdot \pi}{4.25}$

Therefore;

$\displaystyle The \ period, \, T = \frac{2 \cdot \pi}{\left(\frac{2 \cdot \pi}{4.25} \right)} = 4.25$

The time it takes Emily to make one rotation, T = 4.25 minutes

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