SME
Grade 11 Math · Problem type
Model a turning wheel
Use this for “A Ferris wheel is 42 m at the top, 2 m at the bottom and turns once in 8 min. How high is a rider after 2 min?”
Chart
Midline y = 22 m
Amplitude 20 m
Period 8 min
h(2) = 22
This is an example. Type your own values to start a new problem.
top
Top height
example · m
bottom
Bottom height
example · m
T
Time per turn
example · min
A
Amplitude
worked out (not typed) · m
k
Midline height
worked out (not typed) · m
B
Radians per minute
worked out (not typed)
H
Time to the top
worked out (not typed) · min
t
Time
example · min
h
Height
worked out (not typed) · m
Formulas
A
=
top − bottom
2
20
=
42 − 2
2
k
=
top + bottom
2
22
=
42 + 2
2
B
=
2π
T
π
4
=
2π
8
H
=
T
2
4
=
8
2
h
=
k
−
A
×
cos(
B
×
t
)
22
=
22
−
20
×
cos(
π
4
×
2)
Assumptions
•
The rider starts at the bottom, so h = k − A cos(Bt): the cosine flipped.
•
The midline k is halfway between top and bottom; A is half the distance between them.
•
One turn takes T minutes, so B = 2π ÷ T; the top comes at H = T ÷ 2.
Step-by-step
WE KNOW
top = 42 m, bottom = 2 m, T = 8 min, t = 2 min
FIND
amplitude (A), midline height (k), radians per minute (B), time to the top (H), height (h)
Step 1 · Find amplitude (A)
Use A = (top − bottom) ÷ 2
The radius: half the distance from bottom to top.
A
=
42 − 2
2
A
=
40
2
A
=
20
m
Step 2 · Find midline height (k)
Use k = (top + bottom) ÷ 2
The height of the center: halfway.
k
=
42 + 2
2
k
=
44
2
k
=
22
m
Step 3 · Find radians per minute (B)
Use B = 2π ÷ T
One
turn
is
2π
radians
in
T
minutes.
B
=
2π
8
B
=
π
4
(≈
0.79)
Step 4 · Find time to the top (H)
Use H = T ÷ 2
Half a turn from the bottom is the top.
H
=
8
2
H
=
4
min
Step 5 · Find height (h)
Use h = k − A × cos(B × t)
At
t
=
0
the
cosine
is
1,
so
the
rider
starts
at
k
−
A
,
the
bottom.
h
=
22
−
20
×
cos(
π
4
×
2)
h
=
22
−
20
×
cos(
π
2
)
h
=
22
−
20
×
0
h
=
22
m
Check
Put the numbers back in. Both sides should match.
20
=
42 − 2
2
✓
22
=
42 + 2
2
✓
π
4
=
2π
8
✓
4
=
8
2
✓
22
=
22
−
20
×
cos(
π
4
×
2)
✓
Topic check
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