Chapter 1

Fish Population \(\quad\) A large pond is stocked with fish. The fish population \(P\) is modeled by the formula \(P=3 t+10 \sqrt{t}+140,\) where \(t\) is the number of days since the fish were first introduced into the pond. How many days will it take for the fish population to reach 500\(?\)

\(71-78\) Find the solution of the equation correct to two decimals. $$ 3.02 x+1.48=10.92 $$

Suppose that \(a, b, c,\) and \(d\) are positive numbers such that $$ \frac{a}{b}<\frac{c}{d} $$ show that $$ \frac{a}{b}<\frac{a+c}{b+d}<\frac{c}{d} $$

Recall that the symbol \(\overline{z}\) represents the complex con- jugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. \(z-\overline{z}\) is a pure imaginary number

Solve the equation for \(x\). \(b x^{2}+2 x+\frac{1}{b}=0 \quad(b \neq 0)\)

The Lens Equation If \(F\) is the focal length of a convex lens and an object is placed at a distance \(x\) from the lens, then its image will be at a distance \(y\) from the lens, where \(F, x,\) and \(y\) are related by the lens equation $$ \frac{1}{F}=\frac{1}{x}+\frac{1}{y} $$ Suppose that a lens has a focal length of \(4.8 \mathrm{cm},\) and that the image of an object is 4 \(\mathrm{cm}\) closer to the lens than the object itself. How far from the lens is the object?

\(71-78\) Find the solution of the equation correct to two decimals. $$ 8.36-0.95 x=9.97 $$

Recall that the symbol \(\overline{z}\) represents the complex con- jugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. \(z \cdot \overline{z}\) is a real number

Find all values of \(k\) that ensure that the given equation has exactly one solution. \(4 x^{2}+k x+25=0\)

Dimensions of a Box A large plywood box has a volume of 180 \(\mathrm{ft}^{3}\) . Its length is 9 \(\mathrm{ft}\) greater than its height, and its width is 4 \(\mathrm{ft}\) less than its height. What are the dimensions of the box?

\(71-78\) Find the solution of the equation correct to two decimals. $$ 2.15 x-4.63=x+1.19 $$

Temperature Scales What interval on the Celsius scale corresponds to the temperature range \(50 \leq F \leq 95 ?\)

Recall that the symbol \(\overline{z}\) represents the complex con- jugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. \(z=\overline{z}\) if and only if \(z\) is real

Find all values of \(k\) that ensure that the given equation has exactly one solution. \(k x^{2}+36 x+k=0\)

Radius of a Sphere \(\quad\) A jeweler has three small solid spheres made of gold, of radius \(2 \mathrm{mm}, 3 \mathrm{mm},\) and 4 \(\mathrm{mm}\) . He decides to melt these down and make just one sphere out of them. What will the radius of this larger sphere be?

\(71-78\) Find the solution of the equation correct to two decimals. $$ 3.95-x=2.32 x+2.00 $$

Car Rental Cost \(\quad\) A car rental company offers two plans for renting a car. Plan A: \(\$ 30\) per day and 10\(\notin\) per mile Plan B: \(\$ 50\) per day with free unlimited mileage For what range of miles will plan \(B\) save you money?

Complex Conjugate Roots Suppose that the equation \(a x^{2}+b x+c=0\) has real coefficients and complex roots. Why must the roots be complex conjugates of each other? (Think about how you would find the roots using the quadratic formula.)

Find two numbers whose sum is 55 and whose product is \(684 .\)

Construction Costs The town of Foxton lies 10 \(\mathrm{mi}\) north of an abandoned east-west road that runs through Grimley, as shown in the figure. The point on the abandoned road closest to Foxton is 40 \(\mathrm{mi}\) from Grimley. County officials are about to build a new road connecting the two towns. They have determined that restoring the old road would cost \(\$ 100,000\) per mile, while building a new road would cost \(\$ 200,000\) per mile. How much of the abandoned road should be used (as indicated in the figure) if the officials intend to spend exactly \(\$ 6.8\) million? Would it cost less than this amount to build a new road connecting the towns directly?

\(71-78\) Find the solution of the equation correct to two decimals. $$ 3.16(x+4.63)=4.19(x-7.24) $$

Long-Distance Cost A telephone company offers two long-distance plans. Plan A: \(\$ 25\) per month and 5\(\phi\) per minute Plan B: \(\$ 5\) per month and 12\(\notin\) per minute. For how many minutes of long-distance calls would plan \(\mathrm{B}\) be financially advantageous?

The sum of the squares of two consecutive even integers is 1252 . Find the integers.

Powers of \(i\) Calculate the first 12 powers of \(i,\) that is, \(i, i^{2}, i^{3}, \ldots, i^{12} .\) Do you notice a pattern? Explain how you would calculate any whole number power of \(i,\) using the pattern you have discovered. Use this procedure to calculate \(i^{446} .\)

Distance, Speed, and Time A boardwalk is parallel to and 210 ft inland from a straight shoreline. A sandy beach lies between the boardwalk, and the shoreline. A man is standing on the boardwalk, exactly 750 ft across the sand from his beach umbrella, which is right at the shoreline. The man walks 4 \(\mathrm{ft} / \mathrm{s}\) on the boardwalk and 2 \(\mathrm{ft} / \mathrm{s}\) on the sand. How far should he walk on the boardwalk before veering off onto the sand if he wishes to reach his umbrella in exactly 4 \(\min 45 \mathrm{s} ?\)

\(71-78\) Find the solution of the equation correct to two decimals. $$ 2.14(x-4.06)=2.27-0.11 x $$

Driving Cost It is estimated that the annual cost of driving a certain new car is given by the formula $$ C=0.35 m+2200 $$ where \(m\) represents the number of miles driven per year and \(C\) is the cost in dollars. Jane has purchased such a car, and decides to budget between \(\$ 6400\) and \(\$ 7100\) for next year's driving costs. What is the corresponding range of miles that she can drive her new car?

A rectangular garden is 10 \(\mathrm{ft}\) longer than it is wide. Its area is 875 \(\mathrm{ft}^{2} .\) What are its dimensions?

Volume of Grain Grain is falling from a chute onto the ground, forming a conical pile whose diameter is always three times its height. How high is the pile (to the nearest hundredth of a foot) when it contains 1000 \(\mathrm{ft}^{3}\) of grain?

\(71-78\) Find the solution of the equation correct to two decimals. $$ \frac{0.26 x-1.94}{3.03-2.44 x}=1.76 $$

Air Temperature As dry air moves upward, it expands and in so doing cools at a rate of about \(1^{\circ} \mathrm{C}\) for each 100 -meter rise, up to about 12 \(\mathrm{km} .\) (a) If the ground temperature is \(20^{\circ} \mathrm{C},\) write a formula for the temperature at height \(h .\) (b) What range of temperatures can be expected if a plane takes off and reaches a maximum height of 5 \(\mathrm{km} ?\)

A rectangular bedroom is 7 \(\mathrm{ft}\) longer than it is wide. Its area is 228 \(\mathrm{ft}^{2} .\) What is the width of the room?

Radius of a Tank A spherical tank has a capacity of 750 gallons. Using the fact that one gallon is about \(0.1337 \mathrm{ft}^{3},\) find the radius of the tank (to the nearest hundredth of a foot).

\(71-78\) Find the solution of the equation correct to two decimals. $$ \frac{1.73 x}{2.12+x}=1.51 $$

Airline Ticket Price A charter airline finds that on its Saturday flights from Philadelphia to London, all 120 seats will be sold if the ticket price is \(\$ 200 .\) However, for each \(\$ 3\) increase in ticket price, the number of seats sold decreases by one. (a) Find a formula for the number of seats sold if the ticket price is \(P\) dollars. (b) Over a certain period, the number of seats sold for this flight ranged between 90 and \(115 .\) What was the corresponding range of ticket prices?

A farmer has a rectangular garden plot surrounded by 200 \(\mathrm{ft}\) of fence. Find the length and width of the garden if its area is 2400 \(\mathrm{ft}^{2}\) .

Dimensions of a Lot \(\quad\) A city lot has the shape of a right triangle whose hypotenuse is 7 ft longer than one of the other sides. The perimeter of the lot is 392 ft. How long is each side of the lot?

\(79-92\) Solve the equation for the indicated variable. $$ P V=n R T ; \quad \text { for } R $$

Accuracy of a Scale \(\quad\) A coffee merchant sells a customer 3 lb of Hawaiian Kona at \(\$ 6.50\) per pound. His scale is accurate to within \(\pm 0.03\) lb. By how much could the cus- tomer have been overcharged or undercharged because of possible inaccuracy in the scale?

TV Monitors Two television monitors sitting beside each other on a shelf in an appliance store have the same screen height. One has a conventional screen, which is 5 in. wider than it is high. The other has a wider, high-definition screen, which is 1.8 times as wide as it is high. The diagonal measure of the wider screen is 14 in. more than the diagonal measure of the smaller. What is the height of the screens, correct to the nearest 0.1 in?

\(79-92\) Solve the equation for the indicated variable. $$ F=G \frac{m M}{r^{2}} ; \quad \text { for } m $$

Gravity The gravitational force \(F\) exerted by the earth on an object having a mass of 100 \(\mathrm{kg}\) is given by the equation $$ F=\frac{4,000,000}{d^{2}} $$ where \(d\) is the distance (in \(\mathrm{km} )\) of the object from the center of the earth, and the force \(F\) is measured in newtons (N). For what distances will the gravitational force exerted by the earth on this object be between 0.0004 \(\mathrm{N}\) and 0.01 \(\mathrm{N} ?\)

Depth of a Well One method for determining the depth of a well is to drop a stone into it and then measure the time it takes until the splash is heard. If \(d\) is the depth of the well (in feet) and \(t_{1}\) the time (in seconds) it takes for the stone to fall, then \(d=16 t_{1}^{2},\) so \(t_{1}=\sqrt{d} / 4 .\) Now if \(t_{2}\) is the time it takes for the sound to travel back up, then \(d=1090 t_{2}\) because the speed of sound is 1090 \(\mathrm{ft} / \mathrm{s}\) . So \(t_{2}=d / 1090 .\) Thus, the total time elapsed between dropping the stone and hearing the splash is $$ t_{1}+t_{2}=\sqrt{d} / 4+d / 1090 $$ How deep is the well if this total time is 3 s?

\(79-92\) Solve the equation for the indicated variable. $$ \frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}} ; \quad \text { for } R_{1} $$

Bonfire Temperature In the vicinity of a bonfire, the temperature \(T\) in \(^{\circ} \mathrm{C}\) at a distance of \(x\) meters from the center of the fire was given by $$ T=\frac{600,000}{x^{2}+300} $$ At what range of distances from the fire's center was the temperature less than \(500^{\circ} \mathrm{C} ?\)

A small-appliance manufacturer finds that the profit \(P\) (in dollars) generated by producing \(x\) microwave ovens per week is given by the formula \(P=\frac{1}{10} x(300-x)\) provided that \(0 \leq x \leq 200\) . How many ovens must be manufactured in a given week to generate a profit of \(\$ 1250 ?\)

Solving an Equation in Different Ways We have learned several different ways to solve an equation in this section. Some equations can be tackled by more than one method. For example, the equation \(x-\sqrt{x}-2=0\) is of quadratic type: We can solve it by letting \(\sqrt{x}=u\) and \(x=u^{2},\) and factoring. Or we could solve for \(\sqrt{x},\) square each side, and then solve the resulting quadratic equation. Solve the following equations using both methods indicated, and show that you get the same final answers. (a) \(x-\sqrt{x}-2=0\) quadratic type; solve for the radical, and square (b) \(\frac{12}{(x-3)^{2}}+\frac{10}{x-3}+1=0 \quad\) quadratic type; multiply by LCD

\(79-92\) Solve the equation for the indicated variable. $$ P=2 l+2 w ; \quad \text { for } w $$

Falling Ball Using calculus it can be shown that if a ball is thrown upward with an initial velocity of 16 \(\mathrm{ft} / \mathrm{s}\) from the top of a building 128 \(\mathrm{ft}\) high, then its height \(h\) above the ground \(t\) seconds later will be $$ h=128+16 t-16 t^{2} $$ During what time interval will the ball be at least 32 \(\mathrm{ft}\) above the ground?

\(79-92\) Solve the equation for the indicated variable. $$ \frac{a x+b}{c x+d}=2 ; \quad \text { for } x $$