Chapter 4

Lucia has decided to take up swimming. She begins her self-designed swimming program by swimming 20 lengths of a 25 -yard pool. Every 4 days she adds 2 lengths to her workout. Model this situation using a continuous function. In what way is this model not a completely accurate re ection of reality?

A social worker gets paid \(\$ D\) per hour up to 40 hours per week. If he puts in more than 40 hours, the hours over 40 count as overtime, which pays an additional \(50 \%\) per hour. Express his weekly wages as a function of \(x\), where \(x\) is the number of hours he has worked that week. (You ll have to write a function in two pieces since the pay equation is described in two different ways depending upon the value of \(x .\) )

A photocopying shop has a xed cost of operation of \(\$ 6000\) per month. In addition, it costs them \(\$ 0.01\) per page they copy. They charge customers \(\$ 0.07\) per page. (a) Write a formula for \(R(x)\), the shop s monthly revenue from making \(x\) copies. (b) Write a formula for \(C(x)\), the shop \(\mathrm{s}\) monthly costs from making \(x\) copies. (c) Write a formula for \(P(x)\), the shop s monthly pro t (or loss if negative) from making \(x\) copies. (Pro t is computed by subtracting total costs from the total revenue.) (d) How many copies must they make per month in order to break even? (Breaking even means that the pro \(\mathrm{t}\) is zero; the total costs and total revenue are equal.) (e) Sketch \(C(x), R(x)\), and \(P(x)\) on the same set of axes and label the break-even point. (f) Find a formula for \(A(x)\), the shop s average cost per copy. (g) Make a table of \(A(x)\) for \(x=0,1,10,100,1000,10000\). (h) Sketch a graph of \(A(x)\).

Cindy quit her job as a manager in Chicago s corporate world, put on a backpack, and is now traveling around the globe. Upon arrival in Cairo, she spent \(\$ 34\) the rst day, including the cost of an Egyptian visa. Over the course of the next four days, she spent a total of \(\$ 72\) on food, lodging, transportation, museum entry fees, and baksheesh (tips). She is going to the bank to change enough money to last for three more days in Cairo. How much money might she estimate she ll need? Upon what assumptions is this estimate based?

In ation in Turkey has caused prices of small everyday items to be measured in tens of thousands of lira. One day I went to a market and purchased one container of yogurt and two packets of honey for 180,000 Turkish lira. Two days later I returned to the same market and purchased two containers of yogurt and three packets of honey for 310,000 Turkish lira. (a) Assuming that the price remained constant over this two-day period, what is the price of a yogurt? What is the price of a packet of honey? (b) The gures given in this problem are accurate for the summer of 1998 in the town of Iznik, a beautiful, tiny lakeside town founded nearly 3000 years ago. The exchange rate at the time was 258,000 Turkish lira per dollar. Convert the prices of yogurt and honey into dollars.

An item costs \(\$ 1000\) this year. This is a \(10 \%\) increase over the price last year. What was the price last year? (Caution: it was not \(\$ 900\). It would be wise to give last year s price a name like \(x\), or \(P\), or some other labeling of your choice.)

Find the slope of the line through the two points given. (a) \((3,-1),(-2,-3)\) (b) \((\pi, 2 \pi),(0,-\pi)\) (c) \((\sqrt{2}, 3),(\sqrt{2}, 5)\) (d) \((\sqrt{2}, \sqrt{3}),(1, \sqrt{3})\)

It is \(10: 30\) A.M. Over the past half hour six customers have walked into the corner delicatessen. How many people might the owner expect to miss if he were to close the deli to run an errand for the next 15 minutes? Upon what assumption is this based? Suppose that between \(9: 30\) A.M. and \(11: 30\) A.M. he had 24 customers. Is it reasonable to assume that between 11:30 A.M. and 1:30 P.M. he will have 24 more customers? Why or why not?

According to a study done by Chester Kyle, Ph.D. (Long Distance Cycling, Rodale Press, Emmaus, PA, 1993), adding 6 pounds to a bicycle slowed the rider down by 22 seconds on a certain 2 -mile course. Assume that riding the course without the extra weight took \(K\) seconds (actual time not speci ed). (a) Assuming that this relationship is linear, nd an equation for \(T(w)\), the time needed to complete the course as a function of the amount of extra weight added. (b) What is the rate of change of \(T(w) ?\) Interpret this rate of change in practical terms.

On the same set of axes, sketch lines through point \((0,1)\) with the slopes indicated. Label the lines. (a) slope \(=0\) (b) slope \(=\frac{1}{2}\) (c) slope \(=1\) (d) slope \(=2\) (e) slope \(=-\frac{1}{2}\) (f) slope \(=-1\) (g) slope \(=-2\)

A moving company charges a minimum of $$\$250$$ for a move. An additional $$\$ 100$$ per hour is charged for time in excess of two hours. Write a function \(C(t)\) that gives the cost of a move that takes \(t\) hours to complete.

$$ \text { For Problems } 4 \text { through } 13, \text { find the equation of the line with the given characteristics. } $$ $$ \text { Slope }-\frac{1}{2}, \text { passing through }(-2,-3) $$

The three most commonly used temperature scales are the Fahrenheit \(\left({ }^{\circ} \mathrm{F}\right)\), the Celsius \(\left({ }^{\circ} \mathrm{C}\right)\), and the Kelvin (K, an absolute temperature) scales. One interval on the Kelvin scale is equal to one degree Celsius. The freezing point of water is \(0^{\circ} \mathrm{C}\), which is \(32^{\circ} \mathrm{F}\) and \(273.15\) on the Kelvin scale. The boiling point of water is \(100^{\circ} \mathrm{C}\) and \(212^{\circ} \mathrm{F}\). On the Celsius scale, the interval between the freezing and boiling points of water is divided into 100 degrees while on the Fahrenheit scale it is divided into 180 degrees. You have been given more than enough information to answer the following questions! You ll have to select the information you will use. (a) Write a formula for a function that takes as input degrees Celsius and gives as output degrees Fahrenheit. (b) Write a formula for a function that takes as input degrees Fahrenheit and gives as output degrees Celsius. Do this as ef ciently as possible! The function you ve arrived at is the inverse of the function from part (a). (c) Write a formula for a function that takes as input degrees Celsius and gives the temperature on the Kelvin scale as output. (d) Write a formula for a function that takes as input degrees Fahrenheit and gives the temperature on the Kelvin scale as output. Express this function as the composition of two functions from previous parts of this problem.

$$ \text { For Problems } \text { find the equation of the line with the given characteristics. } $$ $$ \text { Slope } \pi \text { , passing through }(3,5) $$

You ve been presented with two different pay plans for the same job. Plan A offers $$\$ 12$$ per hour with overtime (hours above 40 per week) paying time and a half. Plan B offers $$\$ 14$$ per hour with no overtime. Let \(x\) denote the number of hours you work each week. Let \(P_{\mathrm{A}}(x)\) give the weekly pay under plan \(\mathrm{A}\) and \(P_{\mathrm{B}}(x)\) give the weekly pay under plan B. (a) What is the algebraic formula for \(P_{\mathrm{B}}(x) ?\) (b) What is the algebraic formula for \(P_{\mathrm{A}}(x) ?\) Note that you must de ne this function differently for \(x \leq 40\) and for \(x>40\). Check your answer and make sure that the pay for a 50 -hour work week is $$\$ 660 .$$ (c) i. For what value(s) of \(x\) are the two plans equivalent? ii. For what values of \(x\) is plan B better? (Hint: A good problem-solving strategy is to draw a graph so you can really see what is going on.) (d) True or False: i. \(P_{\mathrm{B}}(x+y)=P_{\mathrm{B}}(x)+P_{\mathrm{B}}(y)\) ii. \(P_{\mathrm{A}}(x+y)=P_{\mathrm{A}}(x)+P_{\mathrm{A}}(y)\) (Hint: If you are not sure how to approach a problem, a good strategy frequently used by mathematicians everywhere! is to try a concrete case. If the statement is false for this special case, then you know the statement is de nitely false. If the statement holds for this special case, then the process of working through the special case may help you determine whether the statement holds in general.) Caution: Since the rule for \(P_{\mathrm{A}}(x)\) changes for \(x>40\), you need to check several cases. If any case doesn \(\mathrm{t}\) hold, then the statement is false.

Economists use indifference curves to show all combinations of two goods that give the same ( xed) level of satisfaction to a household. Generally an indifference curve is nonlinear, but for certain combinations of goods it is possible to have a straight-line indifference curve. The following is a linear indifference curve. Let \(R=\) the number of units of item 1 and \(S=\) the number of units of item \(2 .\) (a) Write an equation for the line in terms of \(S, R, a\), and \(c\). (b) Interpret the meanings of the intercepts. (c) Optional (but suggested for those studying economics): Give an example of two items for which the indifference curve could reasonably be linear.

$$ \text { For Problems } \text { find the equation of the line with the given characteristics. } $$ $$ \text { Passing through points }(0, a) \text { and }(b, 0) $$

You ve written a book and have two publishers interested in putting it out. Both publishers anticipate selling the book for \(\$ 20 .\) The rst publisher guarantees you a at sum of \(\$ 8000\) for up to the rst 10,000 copies sold and will pay \(12 \%\) royalty for any copies sold in excess of 10,000 . For instance, if 10,001 copies were sold, you would receive \(\$ 8002.40 .\) The second publisher offers a royalty of \(10 \%\). Let \(x\) be the number of books sold. Let \(A(x)\) give the income under plan \(\mathrm{A}\) and \(B(x)\) give the income under plan \(\mathrm{B}\). (a) What is the algebraic formula for \(A(x)\) ? (b) What is the algebraic formula for \(B(x)\) ? (c) i. For what value(s) of \(x\) are the two plans equivalent? ii. For what values of \(x\) is plan B better? iii. For what values of \(x\) is plan \(\mathrm{A}\) better?

Stories are told about some of the less fair-minded teams of early baseball (e.g., the Baltimore Orioles of the 1890 s) freezing baseballs until shortly before game time so that although the cover would feel normal, the core of the ball would be much colder. Then, they would attempt to introduce these balls into play when the opposing team was at bat, working on the assumption that the frozen balls would not travel as far when hit. Experiments have shown that a ball whose temperature is \(-10^{\circ} \mathrm{F}\) would travel 350 feet after a given swing of the bat, while a ball whose temperature is \(150^{\circ} \mathrm{F}\) would be hit 400 feet by the same swing. Assume this relationship is linear. Let \(B(T)\) be the distance this swing would produce, where \(T\) is the temperature in degrees Fahrenheit. (a) Find an equation for \(B(T)\). (b) What is the \(B\) -intercept? What is its practical meaning? (c) What is the slope of \(B(T) ?\) What is its practical meaning?

$$ \text { For Problems } \text { find the equation of the line with the given characteristics. } $$ $$ \text { Passing through points }(\pi, 3) \text { and }(-\pi, 5) $$

A horseman has some ponies of his own and boards horses for other people. For his own ponies, he orders 9 bales of hay from the supplier. The total number of bales he orders increases linearly with the number of horses he boards. When he boards 6 horses, he orders a total of 36 bales of hay (for these horses and his ponies). Express the number of bales of hay he orders as a function of the number of horses he boards.

$$ \text { For Problems } \text { find the equation of the line with the given characteristics. } $$$$ \text { Passing through point }(\sqrt{3}, \sqrt{2}) \text { and parallel to } 3 x-4 y=7 $$

$$ \text { For Problems } \text { find the equation of the line with the given characteristics. } $$ $$ \text { Passing through the origin and perpendicular to } \pi x-\sqrt{3} y=12 $$

After 3 miles of dif cult climbing in the morning, a group of hikers has reached a plateau and they are con dent they can maintain a steady pace for the next 10 miles. After covering a total of 13 miles, they ll set camp. Twenty minutes after reaching the plateau, they ve covered \(1 \frac{1}{3}\) miles. Express the total daily mileage as a function of \(t .\) where \(t\) is the number of hours spent hiking since they reached the plateau. What is the domain of the function?

$$ \text { For Problems } \text { find the equation of the line with the given characteristics. } $$ $$ x \text { -intercept of } \sqrt{\pi} \text { and parallel to the } y \text { -axis } $$

At 8:00 A.M., a long-distance runner has run 10 miles and is tiring. She runs until \(9: 00\) A.M. but runs more and more slowly throughout the hour. By \(9: 00\) she has run 16 miles. (a) Sketch a possible graph of distance traveled versus time on the interval from \(8: 00\) to \(9: 00\). What are the key characteristics of this graph? (b) Suppose that at 8:00 A.M. she is running at a speed of 9 miles per hour. Find good upper and lower bounds for the total distance she has run by \(8: 30\) A.M. Explain your reasoning with both words and a graph.

$$ \text { For Problems } \text { find the equation of the line with the given characteristics. } $$ $$ \text { Perpendicular to } y-\pi=\pi(x-1) \text { with a } y \text { -intercept of } 3 $$

This problem focuses on the difference between being piecewise linear (made up of straight lines) and being locally linear (being approximately linear when magni ed enough). Consider the functions \(f, g\), and \(h\) below. \(f(x)=|x+2|-3\) \(g(x)=\left\\{\begin{array}{ll}x & \text { for } x \leq 0 \\ x^{2} & \text { for } x>0\end{array}\right.\) \(h(x)=(x-1)^{10}+1\) (a) Graph \(f, g\), and \(h\). (b) Specify all intervals for which the given function is linear (a straight line.) i. \(f\) ii. \(g\) iii. \(h\) (c) Specify the point(s) at which the given function is not locally linear (that is, where it does not look like a straight line, no matter how much you zoom in). i. \(\bar{f}\) ii. \(g\) iii. \(h\)

$$ \text { For Problems } \text { find the equation of the line with the given characteristics. } $$ $$ \text { Horizontal and passing through }\left(-\sqrt{\pi}, \pi^{2}\right) $$

As part of a conservation effort we want to buy a monogrammed mug for every student, staff, and faculty member in the mathematics department. We check with several companies and get the following price quotes. Great Mugs will charge \(\$ 20\) just to place the order and then they charge an additional \(\$ 6\) for each mug that we order. Name It will only charge \(\$ 10\) to process the order and has a varying scale depending upon the number of mugs ordered. For the rst 20 mugs we order, the cost is \(\$ 7\) per mug; for the next 50 mugs, the cost is \(\$ 6\) per mug; and for all mugs after that, the cost is \(\$ 5\) per mug. Let \(G(x)\) be the cost of ordering \(x\) mugs from Great Mugs. Let \(N(x)\) be the cost of ordering \(x\) mugs from Name It. (a) Graph \(G(x)\) and \(N(x)\). (b) Write functions for \(G(x)\) and \(N(x)\). (c) For which values of \(x\) is it cheaper to order from Great Mugs as compared to ordering from Name It? (d) How much can the difference in prices between the two companies ever be if we place an order for the same number of mugs from each company?

$$ \text { For Problems } \text { find the equation of the line with the given characteristics. } $$ $$ \text { Vertical and passing through }\left(-\sqrt{\pi}, \pi^{2}\right) $$

For Problems 14 through 16, find the slope of the secant line passing through points \(P\) and \(Q\), where \(P\) and \(Q\) are points of the graph of \(f(x)\) with the indicated \(x\) -coordinates. \(f(x)=\frac{3}{x}+2 x ;\) the \(x\) -coordinates of \(P\) and \(Q\) are \(x=3\) and \(x=3+k(k \neq 0)\), respectively.

find the slope of the secant line passing through points \(P\) and \(Q\), where \(P\) and \(Q\) are points of the graph of \(f(x)\) with the indicated \(x\) -coordinates. \(f(x)=x^{2}+3 x+1 ;\) the \(x\) -coordinates of \(P\) and \(Q\) are \(x=b\) and \(x=b+h(h \neq 0)\), respectively.

find the slope of the secant line passing through points \(P\) and \(Q\), where \(P\) and \(Q\) are points of the graph of \(f(x)\) with the indicated \(x\) -coordinates. \(f(x)=a x^{2}+b x+c ;\) the \(x\) -coordinates of \(P\) and \(Q\) are \(x=k\) and \(x=k+h(h \neq 0)\), respectively.

There is a proliferation of telephone-call billing schemes. According to one scheme, a call to anywhere in the United States is billed at 50 cents for the rst three minutes and \(9.8\) cents per minute after that. Express the cost of a call as a function of its duration.nates.

From the early \(1500 \mathrm{~s}\) to nearly 1700, the Turkish town of Iznik was famous for its beautiful colored tiles. In the 1990 s, tile-making was pursued with renewed vigor in the town. In the late \(1990 \mathrm{~s}\), a new mosque was built, and the walls, both inside and outside, are currently being covered with the blue and red tiles for which the town is known. If the mosque cost \(C\) dollars to construct with an additional \(T\) dollars for each tile used, nd the total cost as a function of \(x\), where \(x\) is the number of tiles used.