I don't know how to find the equation. Help me, please!

Question

I am riding a Ferris wheel. At time t = 10 seconds I am at its highest point of 60 feet above the ground. At t = 190 seconds the wheel returns me to this highest point. If the radius of the Ferris wheel is 25 feet, answer the following:

a. What is the equation for my height at any time t?

b. How high above the ground was I at time t = 45 seconds?

c. At what times from 0 to 190 seconds will I be 30 feet above the ground? (Solve graphically)

Answer 1

Solution: Using radians

Use Parent function: y = cos(x)

Horizontal Stretch Factor:

Period = 190 - 10 = 180

k(180) = 2pi

k = pi/90

Horizontal shift:

It takes 10 seconds to get to the top.(maximum)

So, horizontal shift = 10 to the right

Vertical shift;

Diameter = 2*radius = 2*25 = 50

When at top, h = 60

Bottom of Ferris wheel is 10 feet off the ground

So, Center of Ferris wheel is 35 feet above ground

Vertical shift = 35

Vertical stretch = radius = 25

y = height

y = 25*cos[ k(x - 10) ] + 35

y = 25*cos[ (pi/90)(x - 10) ] + 35

On calculator:

y1 = 25*cos[ (pi/90)(x - 10) ] + 35

y2 = 30

ANSWERS:

(a) y = 25*cos[ (pi/90)(x - 10) ] + 35

(b) Use TRACE on calculator. Enter x = 45 to get y = 43.551 feet

(c) Find two intersections of y1 and y2 occur at x = 60.768, 139.23 seconds