For the body to function properly, the heart needs to pump blood at a sufficient rate to maintain an adequate and continuous supply of oxygen and other nutrients to the brain and other vital organs.

Cardiac output$($CO$)$ is the term that describes the amount of blood your heart pumps each minute where the units are L$/$min$.$ It is a vital measure of the heart's efficiency in pumping blood which is essential for diagnosing heart conditions, monitoring critically ill patients, and guiding treatment decisions. A common method used by researchers to measure cardiac output is known as the Dye Dilution Method.

A known quantity of dye is typically injected directly into a central vein, such as the right atrium, and the concentration of the dye is then measured as it passes through a peripheral artery until there is no more dye in the bloodstream. By monitoring the changes in dye concentration over time in the arterial blood, the cardiac output is then measured as follows:

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C O$=\backslash$frac$\{$D$\}\{\backslash$int$\_\{0\}^\{$L$\}$ c$($t$)$ d t$\}$

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where:

$-\backslash (\backslash$quad L $\backslash )$ represents the duartion length at which the dye concentration was measured.

$-\backslash (\backslash$quad D $\backslash )$ represents the known amount of dye used in mg $.$

$-\backslash ($ c$($t$)\backslash )$ represents the dye concentration at time $\backslash ($ t $\backslash )$ in $\backslash (\backslash$mathrm$\{$mg$\}/\backslash$mathrm$\{$L$\}\backslash ).$

In practice, it tends to be unfeasable to continuously measure the dye concentration $\backslash ($ c$($t$)\backslash )$ and so quite often the integral has to be approximated using only a reasonable amount of concentration measuremanets at various times.

Using $16$ mg of dye, a team of researchers measured the dye concentration $\backslash ($ c$($t$)\backslash )$ in a patients peripheral artery in $2$ second intervals for a total of $20$ seconds and the data is presented in the table below.

$($a$)$ Using a left endpoint approximation, approximate the Cardiac Output of this patient. Round your answer to three decimal places. Make sure your answer matches the correct units!

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C O$=$

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$($b$)$ Using a right endpoint approximation, approximate the Cardiac Output of this patient. Round your answer to three decimal places. Make sure your answer matches the correct units!

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$\backslash$mathrm$\{$CO$\}=$

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$($c$)$ Suppose the team of scientists run the expiriment another $20$ times, which of the following statements will be correct.

right endpoint will more often than not lead to better CO approximations will always yeild the same CO approximation.

left endpoint and right endpoint approximations will always yeild the same CO approximation.

left endpoint will more often than not lead to better CO approximations will always yeild the same CO approximation.

None of the above