QUESTIONS

In environmental engineering, groundwater remediation is a crucial process for cleaning up contaminated groundwater. One common remediation technique is pump$-$and$-$treat, where groundwater is pumped out of the subsurface, treated to remove contaminants, and then re$-$injected back into the ground. Suppose you are tasked with designing a pump$-$andtreat system for a contaminated groundwater site. The goal is to determine the optimal pumping rate $(Q)$ that maximizes the removal of contaminants from the groundwater over a certain period of time.

What is the pumping rate $(Q)$ that achieves a contaminant concentration of $C\_$target $=0\mathrm{.}15m\frac{g}{L}$ within a time period of $t=8$ days.

Assume that the contaminant concentration in the groundwater is described by the following equation:

$C=0\mathrm{.}5**\frac{Q^2+Q-2}{t^2+t+0\mathrm{.}1}$

$f(t,Q)=C_t$arget

Tolerance $=0\mathrm{.}001$

a$)(60$ pts$)$ Find the solution with three numerical method

b$)(20$ pts$)$ Draw the corresponding graph$/$s

c$)(20$ pts$)$ Comparison and Discussion

Compare the results obtained from the four methods.

Discuss the convergence and accuracy of each method.

Explain the advantages and disadvantages of each method in the context of this problem.

MATLAB Task: Implement all the methods in MATLAB and provide the code and final root estimate. Please avoid using several files instead one $.$m file for all solutions with relevant sub$-$sections.