Cardiac output is the rate R of volume of blood pumped by the heart per unit time

Question:

Cardiac output is the rate R of volume of blood pumped by the heart per unit time (in liters per minute).

Doctors measure R by injecting A mg of dye into a vein leading into the heart at t = 0 and recording the concentration c(t) of dye (in milligrams per liter) pumped out at short regular time intervals (Figure 8).

Inject dye here Blood flow Measure concentration here c(t) (mg/liter) y = c(t) t(s)

(a) Explain: The quantity of dye pumped out in a small time interval [t, t + Δt] is approximately Rc(t)Δt.

(b) Show that S A = R T c(t) dt, where T is large enough that all of the dye is pumped through the heart but not so large that the dye returns by recirculation.