Question

Assume that the volume of air in an average adult’s lung follows a transformed sine function, as function of time t in seconds,

V (t) = A + B sin(ωt)
Below is a table of measurements of the volume. Identify constants A, B, and ω so that the function above fits these measurements. After 12 seconds, the measurements will repeat.
t - 0 3 6 9 12
V(t) - 7 9 7 5 7

Answer 1

Explanation:

Given that,

v(t) = A + B•sin(ωt)

Then,

When t = 0 v(t)= 7

7 = A + B•Sin(ω×0)

7 = A + B•Sin0

7 = A

Then,

A = 7

v = 7 + B•sin(ωt)

So,

When t = 3, v(t) = 9

v = A + B•sin(ωt)

9 = 7 + B•Sin(3w)

9-7 = B•sin(3ω)

B•sin(3w) = 2. Equation 1

Also, at t = 6 v(t) = 7, at this point, when it returns back to7, it has complete one oscillation

v = A + B•sin(ωt)

7 = 7 + B•Sin(6w)

7-7 = B•sin(6ω)

B•sin(6w) = 0

Sin(6w) = 0 / B

Sin(6w) = 0

Take arcsin of both sides

6w = Sin~1(0)

6w = π, since it has complete one oscillation

Then, w = π /6

w = π/6

Then,

v(t) = 7 + B•Sin(πt/6)

From equation 1

B•sin(3w) = 2.

B•Sin(3 × π/6) = 2

B•Sin(½π) = 2

B = 2

Then,

v(t) = A + B•sin(ωt)

A = 7, B = 2 and w = π/6

v(t) = 7 + 2•sin(πt/6)