Chapter 1

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

Radius of a Sphere 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?

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

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 that you have discovered. Use this procedure to calculate \(i^{4446}\)

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

Dimensions of a Room 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?

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?

Find the solution of the equation rounded to two decimals. \(\frac{1.73 x}{2.12+x}=1.51\)

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

Dimensions of a Garden \(A\) 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 \(\mathrm{ft}^{2}\) .

Solve the equation for the indicated variable. \(P V=n R T ; \quad\) for \(R\)

Long-Distance cost A telephone company offers two long-distance plans: Plan A: \(\quad \$ 25\) per month and 5\(€\) per minute Plan B: \(\$ 5\) per month and 12\(€\) per minute For how many minutes of long-distance calls would Plan B be financially advantageous?

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 sandy beach lies on the boardwalk, exactly 750 \(\mathrm{ft}\) 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 \(\min 45 \mathrm{s} ?\)

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

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?

Dimensions of a Lot 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?

Solve the equation for the indicated variable. \(P=2 l+2 w ; \quad\) for \(w\)

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 an airplane takes off and reaches a maximum height of 5 \(\mathrm{km} ?\)

Profit 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 \(\$ 1250 ?\)

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, rounded to the nearest 0.1 in.?

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

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?

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 \(\mathrm{s} ?\) (See the following figure.)

Solve the equation for the indicated variable. \(\frac{a x+b}{c x+d}=2 ;\) for \(x\)

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

Dimensions of a Can A cylindrical can has a volume of 40\(\pi \mathrm{cm}^{3}\) and is 10 \(\mathrm{cm}\) tall. What is its diameter? [Hint: Use the volume formula listed on the inside front cover of this book.]

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, the equating \(\sqrt{x}=u\) 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:

Solve the equation for the indicated variable. \(a-2[b-3(c-x)]=6 ; \quad\) for \(x\)

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 \(\mathrm{N} ?\)

Dimensions of a Lot A parcel of land is 6 \(\mathrm{ft}\) longer than it is wide. Each diagonal from one corner to the opposite corner is 174 \(\mathrm{ft}\) long. What are the dimensions of the parcel?

Solve the equation for the indicated variable. \(a^{2} X+(a-1)=(a+1) x, \quad\) for \(x\)

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 \(500^{\circ} \mathrm{C} ?\)

Height of a Flagpole A flagpole is secured on opposite sides by two guy wires, each of which is 5 \(\mathrm{ft}\) longer than the pole. The distance between the points where the wires are fixed to the ground is equal to the length of one guy wire. How tall is the flagpole (to the nearest inch)?

Solve the equation for the indicated variable. \(\frac{a+1}{b}=\frac{a-1}{b}+\frac{b+1}{a} ; \quad\) for \(a\)

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?

Distance, Speed, and Time A salesman drives from Ajax to Barrington, a distance of \(120 \mathrm{mi},\) at a steady speed. He then increases his speed by 10 \(\mathrm{mi} / \mathrm{h}\) to drive the 150 \(\mathrm{mi}\) from Barrington to Collins. If the second leg of his trip took 6 \(\mathrm{min}\) more time than the first leg, how fast was he driving between Ajax and Barrington?

Solve the equation for the indicated variable. \(V=\frac{1}{3} \pi r^{2} h, \quad\) for \(r\)

Gas Mileage The gas mileage \(g\) (measured in mi/gal) for a particular vehicle, driven at \(v\) mi/h, is given by the formula \(g=10+0.9 v-0.01 v^{2},\) as long as \(v\) is between 10 \(\mathrm{mi} / \mathrm{h}\) and 75 \(\mathrm{mi} / \mathrm{h}\) . For what range of speeds is the vehicle's mileage 30 \(\mathrm{mi} / \mathrm{gal}\) or better?

Distance, Speed, and Time Kiran drove from Tortula to Cactus, a distance of 250 \(\mathrm{mi}\) . She increased her speed by 10 \(\mathrm{mi} / \mathrm{h}\) for the \(360-\mathrm{mi}\) trip from Cactus to Dry Junction. If the total trip took 11 \(\mathrm{h}\) , what was her speed from Tortula to Cactus?

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

Stopping Distance For a certain model of car the distance \(d\) required to stop the vehicle if it is traveling at \(v \mathrm{mi} / \mathrm{h}\) is given by the formula $$ d=v+\frac{v^{2}}{20} $$ where \(d\) is measured in feet. Kerry wants her stopping distance not to exceed 240 \(\mathrm{ft}\) . At what range of speeds can she travel?

Distance, Speed, and Time It took a crew 2 h 40 min to row 6 \(\mathrm{km}\) upstream and back again. If the rate of flow of the stream was 3 \(\mathrm{km} / \mathrm{h}\) , what was the rowing speed of the crew in still water?

Solve the equation for the indicated variable. \(a^{2}+b^{2}=c^{2} ; \quad\) for \(b\)

Manufacturer's Profit If a manufacturer sells \(X\) units of a certain product, revenue \(R\) and cost \(C\) (in dollars) are given by $$ \begin{aligned} R &=20 x \\ C &=2000+8 x+0.0025 x^{2} \\ \text { Use the fact that } & \text { profit }=\text { revenue }-\text { cost } \end{aligned} $$ to determine how many units the manufacturer should sell to enjoy a profit of at least $2400.

Speed of a Boat Two fishing boats depart a harbor at the same time, one traveling east, the other south. (See the following figure.) The eastbound boat travels at a speed 3 \(\mathrm{mi} / \mathrm{h}\) faster than the southbound boat. After two hours the boats are 30 \(\mathrm{mi}\) apart. Find the speed of the southbound boat.

Solve the equation for the indicated variable. \(A=P\left(1+\frac{i}{100}\right)^{2} ; \quad\) for \(i\)

Fencing a Garden A determined gardener has 120 \(\mathrm{ft}\) of deer-resistant fence. She wants to enclose a rectangular vegetable garden in her backyard, and she wants the area that is enclosed to be at least \(800 \mathrm{ft}^{2} .\) What range of values is possible for the length of her garden?

Falling-Body Problems Suppose an object is dropped from a height \(h_{0}\) above the ground. Then its height after \(t\) seconds is given by \(h=-16 t^{2}+h_{0}\) , where \(h\) is measured in feet. Use this information to solve the problem. If a ball is dropped from 288 \(\mathrm{ft}\) above the ground, how long does it take to reach ground level?

Solve the equation for the indicated variable. \(V=\frac{4}{3} \pi r^{3} ; \quad\) for \(r\)

Do Powers Preserve Order? If \(a