Question

Hugo's bike tire has a piece of gum stuck to it.

The distance
G
(
t
)
G(t)G, left parenthesis, t, right parenthesis (in
cm
cmstart text, c, m, end text) between the gum and the sidewalk as a function of time
t
tt (in seconds) can be modeled by a sinusoidal expression of the form
a
⋅
sin
⁡
(
b
⋅
t
)
+
d
a⋅sin(b⋅t)+da, dot, sine, left parenthesis, b, dot, t, right parenthesis, plus, d.
At
t
=
0
t=0t, equals, 0, the gum is halfway between the ground and its maximum height, at
35
cm
35 cm35, start text, space, c, m, end text. The gum reaches its maximum height of
70
cm
70 cm70, start text, space, c, m, end text from the ground
π
20
20
π
​
start fraction, pi, divided by, 20, end fraction seconds later.

Answer 1

G(t) = 35*sin(10*t) + 35

Explanation:

In the figure attached, the question is shown. The function has the form:

G(t) = a*sin(b*t) + d

We know that at t = 0, G(t) = 35. Replacing into the equation:

35 = a*sin(b*0) + d

d = 35

We know that at t = π/20, G(t) = 70, which is a maximum. The maximum is reached when sin(b*t) = 1, replacing into the equation:

70 = a*1 + 35

a = 35

If sin(b*t) = 1, then:

b*t = π/2

b*π/20 = π/2

b = 20/2 = 10