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\begin{document}
\noindent
\hfill{\small \ Winter 2022\ \
\noindent {\bf {\large MATH 120 \ \ Final Exam}\\
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Final Exam: 5:00-7:50pm, Saturday March 12, @ARC 147 }\\\\
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%{\small BA: 10:30-11:20am @SMI 405; \ \ BB: TuTh 11:30am-12:20pm @SMI 405}}\\ \\\\
Name: \frame{\rule{0pt}{1.5cm}\rule{8cm}{0pt}} \ \ \ \ \ Quiz section: \frame{\rule{0pt}{1cm}\rule{3cm}{0pt}} \\
7-digit UW ID: \frame{\rule{0pt}{1cm}\rule{5cm}{0pt}} \\\\
\noindent
{\bf Exam Instruction:}
\begin{itemize}
\item
You have 170 minutes to complete 7 questions. Distribute your time wisely.
\item
Show your work to earn full credit. If you can not completely solve a problem, providing reasonable work and steps may earn you some partial credit.
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Box \boxed{\textup{your final answer}} for each problem.
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The last two pages are scratch papers, tear them off and do NOT turn in unless you have written down additional work to be graded.
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{\bf Do NOT write within 1 cm of the edge.} Your exam will be scanned for grading. If you run out of space, specify ``see scratch paper", then write you additional work on the scratch paper and turn it in together with your exam.
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%Leave your answer in the exact form rather than a decimal approximation. For example, you may leave your answer as a fraction, a square root expression, an inverse trig function of an expression, etc.
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You can prepare one hand-written double-sided $8.5"\times11"$ page of notes and bring it to the exam.
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You may use a basic calculator that can NOT graph or solve equation (e.g., TI-30X IIS). All other electronic devices (e.g., cell phone, earbuds) should be turned off and put away during the exam.
\item You must finish the exam independently. {\bf Giving or receiving any assistance on the exam is considered cheating, which will result in a grade of zero for the exam.} Do not discuss the exam questions with other students before the exam grade is released.\\
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%Please write your name AND your 7-digit UW ID number on the upper right corner of the first page of your work. If you don't know your 7-digit student number, please go to MyUW$\to$Unofficial Transcript (under the Academics block). Your student number appears in the upper left of the Transcript page.
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%Receive help from anyone else.
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%Share your work with your classmate.
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%Search or post the exam questions online to get help.
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%Use an online calculator/tool to graph or perform calculus calculation.
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%\item During the exam, I will be holding a live zoom meeting (our regular class zoom link), joining the zoom meeting is encouraged but is not mandatory. You do not have to turn your camera on during the exam. The purpose of the zoom meeting is to provide an environment similar to an actual exam, you may ask questions via chat, and I can try to help with any technical issues occurred during the exam submission.
%\item No late submission will be accepted. Be prepared to stop working on the exam questions at least 10 minutes before the exam ending time so you have enough time to take pictures, convert them to one single PDF file, and upload the file. If you encountered any technical difficulty uploading your file, please contact me through zoom chat or email at least 5 minutes before the exam ending time to receive permission to send me the PDF file right away.
\end{itemize}
\pagebreak
\begin{enumerate}
\item
(14 pts) When a cherry farm has 150 cherry trees planted, each tree yields 40 pounds of cherries per year. For every 10 additional trees planted in the farm, the annual yield per tree will decrease by 2 pounds due to overcrowding.
(a) Let $x$ be the number of trees planted in the farm. Write the annual yield of cherries per tree (in pound) as a linear function of $x$.
\vspace{4cm}
(b) Find the function $f(x)$ that represents the total annual yield from all the cherry trees in the farm. Then find the number of trees $x=c$ that maximizes the total annual yield $f(x)$.
\vspace{6cm}
(c) Find the inverse function of $f(x)$ restricted on the domain $c\leq x \leq 250$ where $c$ is the number you found in part (b). Specify the domain and range of $f^{-1}(x)$.
\vspace{2cm}
\pagebreak
\item
(16 pts) In 2010, there were 30 raccoons (trash pandas) in a neighborhood. The raccoon population grew exponentially to reach 70 in 2016. Then between 2016 and 2020, the raccoon population increased by $12\%$ every year.
In 2010, there were 20 trash cans in the neighborhood. Between 2010 and 2016, the number doubled every three years. Then between 2016 and 2020, the number of trash cans stayed unchanged.
Let $t$ be the number of years since 2010.
(a) Express the number of raccoons between 2010 and 2020 as a piecewise (multi-part) function of $t$.
\vspace{5cm}
(b) Express the number of trash cans between 2010 and 2020 as a piecewise (multi-part) function of $t$.
\vspace{4cm}
(c) Between the year 2010 and 2020, find all the time $t$ when there were the same number of raccoons and trash cans. {\bf Leave your answers in exact form, do *not* use calculator to round your answers to decimal numbers.}
\vspace{3cm}
\pagebreak
\item
(15 pts) You have 100 meters of fencing materials to enclose two identical regions of circular wedge as illustrated in the picture below. Find the radius $r$ (in meter) and the angle $\theta$ (in radiance) of the circular wedge so that the total fenced area is maximized?
({\bf Note that the line of radius shared by the two circular wedges *is* part of the fence. The total fenced area is referring to the total area of two identical circular wedges. })
\includegraphics[scale=0.4]{f1}
\vspace{10cm}
\pagebreak
\item
%Given the following information of a linear-to-linear rational function $f(x)=\df{ax+b}{cx+d}$:
%
%\begin{itemize}
%\item
%$x$--intercept is $(-2,0)$, $y$--intercept is $\left(0, \frac{4}{3}\right)$.
%\item
%vertical asymptote is $x=-3$; horizontal asymptote is $y=2$.
%
%
%\end{itemize}
(15 pts) Consider the linear-to-linear rational function $f(x)=\df{4x-6}{x-2}$:
(a) Identify the $x$--intercept, $y$--intercept, vertical asymptote and horizontal asymptote of $f(x)$.
\vspace{5cm}
(b) Briefly sketch the graph of $f(x)$ on the $xy$--plane below, your graph should address what you found in (a).
\includegraphics[scale=0.4]{f2}
(c) Sketch the graph of $g(x)=|x+1|$ on the $xy$--plane above. Then solve the equation $f(x)=g(x)$. Leave your answer in exact form.
\pagebreak
\item
(15 pts) A sea lion is chilling on a buoy 2 miles due north and 2 miles due west of a person in a kayak. At noon, the person starts paddling due east at a speed of 3 mph, and the sea lion starts swimming due southeast at a speed of 5 mph. The path of the sea lion makes an angle of $20^{\circ}$ with the path of the person. Impose a coordinate system with the origin being the person's starting position.
For each of the following questions, round the numbers in your final answer to have 1 decimal place.
(a) Find the coordinate of the point of intersection between the path of the kayaking person and the path of the swimming sea lion.
\vspace{5.5cm}
(b) Find the parametric equations for the $x$ and $y$ coordinate of the sea lion $t$ hours after noon.
\vspace{7cm}
(c) Use your answer for part (b) to find the distance between the person and the sea lion at 4 pm. \vspace{5cm}
\pagebreak
\item
(14 pts) A Ferris wheel rotates at a constant speed of 6 feet per second. It takes 0.8 minute to complete one revolution. A coordinate system is imposed so that the center of the wheel is the origin and the wheel is rotating counterclockwise. At $t=0$, a butterfly lands at point $P$ referenced by the angle $\alpha$ in the picture, it reaches the top of the ride in 5 seconds.
For each of the following questions, round the numbers in your final answer to have 2 decimal places.
(a) Find the radius $r$ (in ft). Find the angle $\alpha$ (in radiance).
\includegraphics[scale=0.35]{f3b}
%v=6 ft/s
%
%w=2pi/(0.8*60)=pi/24=0.13
%
%r=v/w=6/(pi/24)=144/pi=45.84
%
%phi=pi/2-5*(pi/24)=7pi/24=0.92
\vspace{1.5cm}
(b) Find the parametric equations $x(t)$ and $y(t)$ for the coordinate of the butterfly $t$ seconds later.
\vspace{5cm}
(c) After 30 seconds, the butterfly flies away along a path tangent to the wheel. Find the equation of the path. \\
\pagebreak
\item
(11 pts) A spacecraft landed on a planet whose temperature fluctuates periodically. 30 hours after the landing, the minimum temperature of $4^{\circ}$F was recorded the first time. 72 hours after the landing, the maximum temperature of $120^{\circ}$F was recorded the first time.
\noindent
(a) Find a sinusoidal function $f(t)=A\sin\left[\df{2\pi}{B}(t-C)\right]+D$ which models the temperature $t$ hours after the landing.
\vspace{8.5cm}
\noindent
(b) Within the first 100 hours after the landing, find all the time $t$ when the temperature is $80^{\circ}$F. Round your answers to have two decimal places. \\\\
\end{enumerate}
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scratch paper
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scratch paper
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