Question

A spring is hanging from a ceiling. The length L(t) (in cm) of the spring as a function of time t (in seconds) can be modeled by a sinusoidal expression of the form a*sin(b*t) +d. At t=0, when the spring is exactly in the middle of its oscillation, its length is 7 cm. After 0.5 seconds the spring reaches its maximum length, which is 12 cm. Find L(t).

Answer 1

L(t) = 5·sin(πt) +7

Explanation:

The middle of the oscillation of the given function occurs when t=0. At that point, ...

L(0) = d = 7

The next maximum of the oscillation occurs when the argument of the sine function is π/2.

b·t = π/2

b = π/(2t) = π/(2·0.5) = π

At that maximum, the length is 12, so we have ...

L(0.5) = a·sin(0.5π) +7 = 12

a = 5

The function L(t) is ...

L(t) = 5·sin(πt) +7