   Chapter 8.4, Problem 22E

Chapter
Section
Textbook Problem

After a 5.5-mg injection of dye, the readings of dye concentration, in mg/L, at two-second intervals are as shown in the table. Use Simpson’s Rule to estimate the cardiac output. To determine

To calculate: The cardiac output using Simpson’s rule.

Explanation

Given information:

The quantity of dye is 5.5mg.

Dye concentration is measured in mg/L.

Number of subintervals is 8.

Time is T=16sec.

Calculation:

The Simpson’s rule is shown below:

abf(x)dxSn=Δx3[f(x0)+4f(x1)+2f(x2)+4f(x3)+...+2f(xn2)+4f(xn1)+f(xn)] (1)

The interval width Δx=ban (2)

Here, the value of the function f(x) at x=0ton is f(x0)tof(xn), the upper limit is b, the lower limit is a, and the number of interval is n.

Modify Equation (1) to calculate the dye concentration as shown below:

abc(t)dtSn=Δt3[c(t0)+4c(t1)+2c(t2)+4c(t3)+...+2c(tn2)+4c(tn1)+c(tn)] (3)

Find the width of the interval:

Substitute 16 for b, 0 for a, and 8 for n in Equation (2).

Δx=1608=2

Thus the interval are, 0, 2 sec, 4 sec, 6 sec, 8 sec, 10 sec, 12 sec, 14 sec, and 16 sec.

Modify Equation (3) for n=8.

S8=Δt3[c(t0)+4c(t1)+2c(t2)+4c(t3)+2c(t4)+4c(t5)+2c(t6)+4c(t7)+c(t8)] (4)

Substitute 0 for t0, 2 sec for t1, 4 sec for t2, 6 sec for t3, 8 sec for t4, 10 sec for t5, 12 sec for t6, 14 sec for t7, and 16 sec for t8 in Equation (4)

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