Expert: Paul Klarreich - 1/9/2012

Question
Hello,

I've been working in this problem since yesterday and I don't known what to do next.

The ferris wheel has a diameter of 50 ft and a height of 4 ft off the ground. The ferris wheel makes 1 revolution every 30 seconds. The overpass is 25 ft tall. Jonathan and Monica are at the highest point on the ride 15 seconds after the wheel begins to turn at full speed. During what interval will Jonathan and Monica be able to see over the 25 ft high overpass?

I believe that I have to write an equation,this is what I have so far.

these below are
the points
         
highest point(0,54)
         25
         (0,29)
         25
lowest point (0,4)

amplitude:25ft
period:30sec
Period=2pi/B  2pi/30=pi/15
formula:y=af(b(x-c))+d     h(t)=25cos((pi/15)*t)+29       


Hope to get some feedback from the EXPERT....thanks

Answer
Questioner:vicky
Country:Texas, United States
Category:Advanced Math
Private:No
Subject:Pre-calculus

Question:

Hello,

I've been working in this problem since yesterday and I don't known what to do next.

The ferris wheel has a diameter of 50 ft and a height of 4 ft off the ground. The ferris wheel makes 1 revolution every 30 seconds. The overpass is 25 ft tall. Jonathan and Monica are at the highest point on the ride 15 seconds after the wheel begins to turn at full speed. During what interval will Jonathan and Monica be able to see over the 25 ft high overpass?

>>>>>> Where is the overpass?  Do you mean, simply, that they must be at least 25 feet high?   

I believe that I have to write an equation,this is what I have so far.

these below are the points
         
highest point(0,54)
         25
         (0,29)
         25
lowest point (0,4)

amplitude:25ft
period:30sec
Period=2pi/B  2pi/30=pi/15
formula:y=af(b(x-c))+d     h(t)=25cos((pi/15)*t)+29       

..................................
Ok, you need a sine-or-cosine curve.  It sounds as if you want:

the low point to occur at t = 0.  Conclude: cosine graph inverted:

y = - A cos(B t) + offset.

the low point is at y = 4.
the diameter is 50, so amplitude is 25, and the offset is 29:

y = - 25 cos (B t) + 29

One revolution takes 30 secs, so when t = 30,  Bt = 2 pi


B(30) = 2 pi

B = pi/15

y = - 25 cos (pi t/15) + 29

So you are doing pretty well, except for that minus.
Now the question is:  For what values of t, is y >= 25?

From the picture, it appears that you want t in [7 to 23], approximately.

But we can do better:

Solve:
 - 25 cos (pi t/15) + 29 = 25

 - 25 cos (pi t/15) = -4

   cos (pi t/15) = 4/25 = 0.16

   pi t/15 = arccos(0.16)

   pi t/15 = 15 arccos(0.16) / pi

Now use your calculator.  Mine gives:

6.7327586482211749545405923416697

And 30 minus that for the other.