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\usepackage{enumerate}
\title{MAT 515 Homework 1}
\author{Moira Chas}
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\begin{enumerate}

\item Programt o solve quadratic equations
\item program remainder.
\item Give algorithm to enter numers in calculator.
\item Polar coordinates in geogebra

\item  (From Model Centered Learning, Edited by Lingguo Bu and Robert Schoen)	The Sears/Willis Tower in Chicago is about 442 meters from its roof to the ground. Mark takes a baseball to the roof, and somehow gets it out of the window with no force imposed on it. Now the ball falls freely toward the ground. Assuming that the air has no significant influence on the baseball and the gravitational acceleration in Chicago is approximately 10 m/s2, Mark wonders, without using calculus: (1) How fast is the ball falling? (2) How does the distance from the roof to the ball change over time?
\item (with a calculator) Choose a three digit number in your calculator. Multiply it by 7, multiply it by 11 and multiply it by thirteen. The result is "math-magic": your original number should reappear twice. Can you explain (mathematically) what happened?  State a conjecture and prove it. 
\item (with your calculator)  Choose a positive number. Compute the square root. Compute the square root of the result... What do you observe? Can 
\item Describe an  strategy so player B wins the following game. 
\begin{enumerate}[(a)]
\item Player A enters a three digit number in the calculator. (at most 900)
\item Player B must reduce the given number in at most 5 steps, using only the four basic operations and digits. Player B is only allowed to use exact divisors.
\end{enumerate}

\item Study the strategy of the following game. 
\begin{enumerate}[(a)]
\item Player A enters a  number, computes a square and tells the result to player B.
\item Player B must find the square root of the number Player A gave him, without using the $\sqrt{.}$ key, recording the number of attempts.
\item They reverse the roles. The player with the smallest number of attempts wins. 
\end{enumerate}

\item Game. Players play by turns. the winner is teh player with highest total of points. In each turn a player,
\begin{enumerate}[(a)]
\item Gets a randon integer $A$ between 0 and 5.
\item Gets a random fraction $B$ from the set $\{\frac{2}{5},\frac{1}{3},\frac{3}{5},\frac{1}{2},\frac{3}{4},\frac{2}{3},\frac{2}{5},\frac{4}{5} \}$.
\item Gets a random operation $O$ from $\{+, \times, \div \}$.
\item Estimates the answer for "$A$ O $B$'' and records the estimate.
\item Uses the calculator to determine the exact value of "$A$ O $B$''.
\item Compares the estimate and the exact value, by computing $E$, the result of  dividing the estimated value by the exact value and rounding the result to the nearses hundred.
\item The player has 
$$
\left\{\begin{array}{ll}
\mbox{10 points } &\mbox{if $E \in [0.95, 1.05]$}\\
\mbox{5 points} &   \mbox{if $E \in (0.9, 0.95) \cup (1.05,1.1)$}\\
\mbox{2 points} &   \mbox{if $E \in [0.85, 0.9] \cup [1.1,1.15]$}\\
\mbox{1 point} &   \mbox{if $E \in (0.8, 0.85) \cup (1.15,1.2)$}\\
\mbox{1 point} &   \mbox{if $E \le 0.8$ or $E \ge 1.2$.}
\end{array}
\right.
$$


\end{enumerate}

\item Find the area between the curves $y=x^2$ and $y=x^3$. (NOte that if drawn in calculator with wrong scale, seems that the curves coincide).

\item An open box is made from a square piece of cardboard by cutting an identical square from each corner to form a box (see Figure~\ref{square}). The original cardboard square measures $18 \times 18$ inches. You have to  find out the size of the square to cut out to obtain a box of the largest possible volume in three different ways:
\begin{enumerate}[a]
\item Try a few examples and estimate the volumexx. Use these examples to estimate the answer.
\item Find an expression for the volume in terms of the size of the cut square. Use this expression to produce a graph. Use the graph to estimate the answer.
\item Find the answer by using calculus.
\item Do a figure of the problem in geogebra and compute the volume in terms of the cut square.
\end{enumerate}




\item Suppose you are given a plastic or cardboard square $\square ABCD$ that is an exact copy of another plastic or cardboard square $\square A'B'C'D'$, with corresponding vertices appropriately labeled. Further suppose that the squares are flipped into the air and land as shown in Figure 2. In this case the midpoints of the paths between corresponding vertices fall on a straight line! Find the probability of this alignment randomly happening when the squares are flipped.
With GeoGebra, students can create simulations of the Flippin Squares task and many other such tasks in the transformational geometry domain. Such simulations can be viewed as empirical models that can be manipulated by students to investigate informally the fundamental notions of transformational geometry and discover patterns before being introduced to the major theorems or even the definitions and properties of various transformations. These simulations help students to see the generality of patterns in a phenomenon and provide compelling evidence that the patterns will continue to hold in the infinite space of unexamined cases. They help students to refine conjectures by ruling out cases and even suggest strategic paths to proving conjectures by revealing invariant relationships in the phenomenon. The Flippin Squares task is part of a larger problem space in which students move back and forth between model- centered investigations and synthetic/analytic formalizations thereby contributing to the pluralistic view of mathematics proposed by Cooney and Wiegel (2003).

\item Given an image and preimage of a triangle, determine transformation.

\item  Find the centre and the radius of a circle such that $P = (1,–1)$ and $Q = (3, 5)$ are on the circle and the circle’s centre lies on the line r, whose equation is: $x + y + 2 = 0.$
\item  A rhombus has two vertices $P = (–2, 1)$ and $Q = (0,–3)$ that form one diagonal of the rhombus. The perimeter is 20 cm. Find the remaining vertices and the area of the rhombus.


\item The authorities of four towns are planning to build an airport that will serve the needs of their citizens. Identify the optimal place for the airport location so that the needs of the four towns are served in a fair way. Send a letter to the Ministry of Transportation explaining and documenting your solution.%model center Chapter 7

\item : A car making company is launching a new solar powered car. Recent market research showed that one hundred people would buy the car for a selling price of €5000. Further, the market research showed that for every 1000$\$$ price increase, people’s interest in buying the car would decrease
by one person. Find the best selling price for the car, as to maximize company’s sales revenue. Send a letter explaining how you solved the problem to the company’s sales manager.%model center Chapter 7

\item \begin{enumerate}[a]
\item Construct a triangle ΔABC in the Geometer’s Sketchpad.
\item Trisect (in congruent segments) each side of the triangle.
\item Choose for each side the first trisection point (clockwise relative to the vertices
A, B and C ) and connect each such point with the opposite vertex.
\item\item Investigate the relation between the area of the central triangle that results from
this construction, and the area of the triangle $\triangle ABC.$
\item  Given an arbitrary triangle ΔABC. Trisect (in congruent segments) each side of the triangle. Connect the trisection points to the opposite vertices. Investigate the relation between the central figure’s area and the area of the original triangle ΔABC.
\item Given an arbitrary triangle ΔABC. Five-sect (in congruent segments) each side of the triangle. Connect the two central points of the five-section points to the opposite vertices. Investigate the relation between the central figure’s area and the area of the original triangle ΔABC.
\item Use the same construction as in 6), but divide each side of the triangle into n parts where n is an odd number larger than 5. Investigate the relation between the central figure’s area and the area of the original triangle ΔABC for each such selected value of n.
\item Repeat the procedures of task 6) and 7) but connect instead the two outermost portioned points for each side with the opposite vertex respectively.


\end{enumerate}
\item Initial task: In rectangle $ABCD$,$AB=5$,and $BC=3$,Points  $F$ and $G$ are on $CD$ so that $DF=1$ and $GC=2$ . Lines $AF$ and $BG$ intersect at $E$. Find the area of $\triangle AEB$.

Later task: Spot’s doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with  a two-yard rope. What is the area, in square yards, of the region outside the doghouse that Spot can reach?

\item Today is Wednesday. What day of the week will it be in 11 days? in 95 days? in 320772 days?

\item There is a jug of wine and a kettle of tea. A spoonful of tea is taken from the kettle and poured into the jug. The mixture is thoroughly stirred and a spoonful of the mixture is taken from the jug and poured into the kettle. Is there more tea in the jug or more wine in the kettle?5

\end{enumerate}



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\label{square}
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