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The monthly profit \(P\) for a widget producer is a function of the number \(n\) of widgets sold. The formula is $$ P=-15+10 n-0.2 n^{2} . $$ Here \(P\) is measured in thousands of dollars, \(n\) is measured in thousands of widgets, and the formula is valid up to a level of 15 thousand widgets sold. a. Make a graph of \(P\) versus \(n\). b. Calculate \(P(1)\) and explain in practical terms what your answer means. c. Is the graph concave up or concave down? Explain in practical terms what this means. d. The break-even point is the sales level at which the profit is 0 . Find the break-even point for this widget producer.

We want to form a rectangular pen of area 100 square feet. One side of the pen is to be formed by an existing building and the other three sides by a fence (see Figure \(2.109\) ). Let \(W\) be the length, in feet, of the sides of the rectangle perpendicular to the building, and let \(L\) be the length, in feet, of the other side. a. Find a formula for the total amount of fence needed in terms of \(W\) and \(L\). b. Express, as an equation involving \(W\) and \(L\), the requirement that the total area formed be 100 square feet. c. Solve the equation you found in part b for \(L\). d. Use your answers to parts a and \(\mathrm{c}\) to find a formula for \(F\), the total amount, in feet, of fence needed, as a function of \(W\) alone. e. Make a graph of \(F\) versus \(W\). f. Determine the dimensions of the rectangle that requires a minimum amount of fence.

The temperature \(C\) of a fresh cup of coffee \(t\) minutes after it is poured is given by \(C=125 e^{-0.03 t}+75\) degrees Fahrenheit \(.\) a. Make a graph of \(C\) versus \(t\). b. The coffee is cool enough to drink when its temperature is 150 degrees. When will the coffee be cool enough to drink? c. What is the temperature of the coffee in the pot? (Note: We are assuming that the coffee pot is being kept hot and is the same temperature as the cup of coffee when it was poured.) d. What is the temperature in the room where you are drinking the coffee? (Hint: If the coffee is left to cool a long time, it will reach room temperature.)

The factorial function occurs often in probability and statistics. For a non- negative integer \(n\), the factorial is denoted \(n\) ! (which is read " \(n\) factorial") and is defined as follows: First, 0! is defined to be 1. Next, if \(n\) is 1 or larger, then \(n\) ! means \(n(n-1)(n-2) \cdots 3 \times\) \(2 \times 1\). Thus \(3 !=3 \times 2 \times 1=6\). Consult the Tech nology Guide to see how to enter the factorial operation on the calculator. In some counting situations, order makes a difference. For example, if we arrange people into a line (first to last), then each different ordering is considered a different arrangement. The number of ways in which you can arrange \(n\) individuals in a line is \(n !\). a. In how many ways can you arrange 5 people in a line? b. How many people will result in more than 1000 possible arrangements for a line? c. Suppose you remember that your four-digit bank card PIN number uses \(7,5,3\), and 1 , but you can't remember in which order they come. How many guesses would you need to ensure that you got the right PIN number? d. There are 52 cards in an ordinary deck of playing cards. How many possible shufflings are there of a deck of cards?

The cost of making a can is determined by how much aluminum \(A\), in square inches, is needed to make it. This in turn depends on the radius \(r\) and the height \(h\) of the can, both measured in inches. You will need some basic facts about cans. See Figure \(2.110\). The surface of a can may be modeled as consisting of three parts: two circles of radius \(r\) and the surface of a cylinder of radius \(r\) and height \(h\). The area of these circles is \(\pi r^{2}\) each, and the area of the surface of the cylinder is \(2 \pi r h\). The volume of the can is the volume of a cylinder of radius \(r\) and height \(h\), which is \(\pi r^{2} h\). In what follows, we assume that the can must hold 15 cubic inches, and we will look at various cans holding the same volume. a. Explain why the height of any can that holds a volume of 15 cubic inches is given by $$ h=\frac{15}{\pi r^{2}} \text {. } $$ b. Make a graph of the height \(h\) as a function of \(r\), and explain what the graph is showing. c. Is there a value of \(r\) that gives the least height \(h\) ? Explain. d. If \(A\) is the amount of aluminum needed to make the can, explain why $$ A=2 \pi r^{2}+2 \pi r h $$ e. Using the formula for \(h\) from part a, explain why we may also write \(A\) as $$ A=2 \pi r^{2}+\frac{30}{r} $$

Ohm's law says that when electric current is flowing across a resistor, then the voltage \(v\), measured in volts, is the product of the current \(i\), measured in amperes, and the resistance \(R\), measured in ohms. That is, \(v=i R\). a. What is the voltage if the current is 20 amperes and the resistance is 15 ohms? b. Find a formula expressing resistance as a function of current and voltage. Use your function to find the resistance if the current is 15 amperes and the voltage is 12 volts. c. Find a formula expressing current as a function of voltage and resistance. Use your function to find the current if the voltage is 6 volts and the resistance is 8 ohms.

The per capita growth rate \(r\) (on an annual basis) of a population of grazing animals is a function of \(V\), the amount of vegetation available. A positive value of \(r\) means that the population is growing, whereas a negative value of \(r\) means that the population is declining. For the red kangaroo of Australia, the relationship has been given \({ }^{22}\) as $$ r=0.4-2 e^{-0.008 v} $$ Here \(V\) is the vegetation biomass, measured in pounds per acre. a. Draw a graph of \(r\) versus \(V\). Include vegetation biomass levels up to 1000 pounds per acre. b. The population size will be stable if the per capita growth rate is zero. At what vegetation level will the population size be stable?

Friction loss in fire hoses: When water flows inside a hose, the contact of the water with the wall of the hose causes a drop in pressure from the pumper to the nozzle. This drop is known as friction loss. Although it has come under criticism for lack of accuracy, the most commonly used method for calculating friction loss for flows under 100 gallons per minute uses what is called the underwriter's formula: $$ F=\left(2\left(\frac{Q}{100}\right)^{2}+\frac{Q}{200}\right)\left(\frac{L}{100}\right)\left(\frac{2.5}{D}\right)^{5} $$ Here \(F\) is the friction loss in pounds per square inch, \(Q\) is the flow rate in gallons per minute, \(L\) is the length of the hose in feet, and \(D\) is the diameter of the hose in inches. a. In a 500 -foot hose of diameter \(1.5\) inches, the friction loss is 96 pounds per square inch. What is the flow rate? b. In a 500 -foot hose, the friction loss is 80 pounds per square inch when water flows at 65 gallons per minute. What is the diameter of the hose? Round your answer to the nearest \(\frac{1}{8}\) inch.