What volume of blood passes along the artery in one second?

Question

An artery has a circular cross section of radius 3 millimeters. The speed at which blood flows along the artery fluctuates as the heart beats. The speed after t seconds is 20+5sin(2πt) meters per second.

What volume of blood passes along the artery in one second?

I keep messing up somewhere in my work and keep getting marked wrong on this question. I think I may be getting the antiderivative wrong.

I would appreciate it if someone could write out the steps/clarify this question so I can see where I'm messing up!

Answer 1

Now the radius of the cross section is 3 mm which is the same as 3*10^-3 m.

The cross sectional area, A = πr² = 9π * 10^-6 m²

 

Now the blood is flowing across this cross section at the rate of v(t) = 20 + 5*sin(2πt) m/sec)

 

The volume flow would be dV/dt = A * v(t) => V = ∫[A * v(t)] dt = A ∫ v(t) dt, which is evaluated for one second.

Assume this time is between t₀ = 0 second and t₁ = 1 second.

 

Now ∫v(t) dt = ∫(20 + 5 * sin(2πt)) dt = 20t - (5/(2π) * cos(2πt) + c

 

At t₁ = 1, V₁ = A *[20 - (5/(2π)*1  + c] m³ and at t0 = 0 , V₀  = A * [0 - (5/(2π) * 1 + c]

 

∴The volume is V = V1 - V0 = A * [20 - (5/(2π) + c + (5/(2π)  - c] = 20A = 180π * 10^-6 m³ = 5.6549  * 10^-4 m³

 

                                                                                                               = 5.6549 * 10^-4 m³ * (10³ mm/m)³

 

                                                                                                               =  5.6549 * 10⁵ mm³