# Dye Dilution; Estimate value of an Integral

• oddjobmj
In summary, the dye dilution method is used to measure cardiac output by modeling the dye concentrations with a given function. The cardiac output can be found by using the formula F=A/\int[c(t) dt]010, where A is the known amount of dye and the integral can be estimated using Riemann sum or Simpson's rule. However, the integral can also be solved analytically, which gives a value of approximately 30, and then dividing it by 0.2 gives a cardiac output of 150 mL/s.
oddjobmj

## Homework Statement

The dye dilution is used to measure cardiac output with 6 mg of dye. The dye concentrations, in mg/L, are modeled by c(t)=20te^(-0.6t), 0 =< t =< 10, where t is measured in seconds. Find the cardiac output.

## Homework Equations

Cardiac output is given by: F=A/$\int$[c(t) dt]010 where the amount of dye A is known and the integral can be approximated from the concentration readings.

## The Attempt at a Solution

In this case A=6 mg and c(t) = 20te^(-0.6t).

I've been trying to estimate with a Riemann sum and/or Simpson's rule but I can't figure out how to get the integral in the correct form to estimate with.

Wolfram alpha spits out 54.5916 using a Riemann sum, but the next problem in this packet (same setup with different dye amount) suggests I use Simpson's rule.

When I plug values of 0->10 into the function I get weird values but I can't integrate it without an estimation technique.

That is pretty easy to integrate directly, isn't it? I assume you are required to integrate numerically as practice, but what do you get for the integral, analytically?

I don't get anything at all like "54..", I get around 30 for the integral and then around 0.2 for the fraction.

## 1. What is dye dilution?

Dye dilution is a method used to estimate the value of an integral in a system. It involves injecting a known amount of dye into the system and measuring the concentration of the dye over time. This data is then used to calculate the value of the integral.

## 2. How does dye dilution work?

In dye dilution, a known amount of dye is injected into the system and mixed with the existing fluid. The concentration of the dye is then measured at specific intervals over time. This data is used to create a concentration-time curve, which can be used to calculate the value of the integral.

## 3. What is the purpose of using dye dilution?

The purpose of dye dilution is to estimate the value of an integral in a system. This can be useful in a variety of scientific fields, including medicine, environmental studies, and engineering. It allows scientists to measure the flow rate of fluids, determine the size of a system, and calculate other important parameters.

## 4. What are the limitations of dye dilution?

One limitation of dye dilution is that it assumes the system is well-mixed, meaning the dye is evenly dispersed throughout the fluid. If there are any areas of stagnant flow or turbulence, the results may be inaccurate. Additionally, the properties of the dye, such as its density and viscosity, can also affect the accuracy of the results.

## 5. Are there any alternatives to dye dilution?

Yes, there are other methods for estimating the value of an integral, such as tracer studies using radioactive or non-radioactive substances. These methods have their own advantages and limitations, and the choice of method will depend on the specific needs and constraints of the experiment or study.

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