Still have math questions?

Q:

A tennis player practices by serving $$100$$ times a day, hitting $$200$$ times a day, and volleying $$50$$ times a day. She wants to increase her serving by $$15 \%$$ and decrease her hitting by $$12 \%$$ . If the number of total swings increases by $$10 \%$$ , how many times will she practice volleying a day?

Q:

Suppose $$H ( x ) = 6 \sqrt { x } - 5$$ . Find two functions $$f$$ and $$g$$ such that $$( f \circ g ) ( x ) = H ( x )$$ .

Neither function can be the identity function.

(There may be more than one correct answer.)

Q:

A blouse that is size $$x$$ in country A is size $$s(x)$$ in country B where $$s ( x ) = x + 2$$ . A blouse that is size x in the country B is $$t(x)$$ in country C,where $$t(x)= x- 1$$.Find a function $$f(x)$$ that will convert blouse size in country A to blouse size in country Z.

$$f ( x ) = \square$$ (Simplify your answer.)

Q:

In how many ways can first, second, and third prizes be awarded in a contest with $$930$$ contestants? Assume there are no ties.

Q:

NASA launches a rocket at $$t = 0$$ seconds. Its height, in meters above sea-level, as a function of time given by $$h ( t ) = 4.9 t ^ { 2 } + 157 t + 199$$ .

Assuming that the rocket will splash down into the ocean, at what time does splashdown occur?

The rocket splashes down after  $$\square$$ seconds.

How high above sea level does the rocket get at its peak?

The rocket peaks at $$\square$$ meters above sea-level.

Q:

NASA launches a rocket at $$t$$ = 0 seconds. Its height, in meters above sea-level, as a function of time is given by $$h ( t ) = 4.9t ^ { 2 } + 220 t + 269$$ . Assuming that the rocket will splash down into the ocean, at what time does splashdown occur?

The rocket splashes down after $$\square$$ seconds.

How high above sea level does the rocket get at its peak?

The rocket peak at $$\square$$ meters above sea level.

Q:

Suppose Set B contains $$100$$ elements and the total number elements in either Set $$A$$ or Set B is $$140$$ . If the Sets A and B have $$15$$ elements in common, how many elements are contained in set $$A$$ ?

Q:

Find the complement of the set given that $$U = \{ x | x \in I$$ and $$- 3 \leq x \leq 7 \} .$$

$$\{ - 3 , - 2,0,5,7 \}$$

Q:

A movie theater has a seating capacity of $$189$$ . The theater charges $$\ 5.00$$ for children, $$\ 7.00$$ for students, and $$\ 12.00$$ for adults. There are half as many adults as there are children. If the total ticket sales was $$\1366$$, How many children, students, and adults attended?

Q:

Solve the logarithmic equation.Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer.

Q:

The exponential model A=3151 e" 4 describes the population,A ,of a country in millions t yeas after 2003 .Use the mode to determine when the population of the county will be 318 million.

Q:

Find the sum.

(a) $$\frac { 1 } { 3 } + \frac { 2 } { 3 ^ { 2 } } + \frac { 2 ^ { 2 } } { 3 ^ { 3 } } + \frac { 2 ^ { 3 } } { 3 ^ { 4 } } + \cdots + \frac { 2 ^ { 9 } } { 3 ^ { 10 } }$$

(b) $$1 + \frac { 1 } { 2 ^ { 1 / 2 } } + \frac { 1 } { 2 } + \frac { 1 } { 2 ^ { 3 / 2 } } + \cdots$$

Q:

Leaning Ladder A $$20 - ft$$ ladder leans against a building so that the angle between the ground and the ladder is $$72 ^ { \circ }$$ . How high does the ladder reach on the building?

Q:

Land Value Land in downtown Columbia is valued at $$\ 20$$ a square foot. What is the value of a triangular lot with sides of lengths $$112,148$$ , and $$190 ft$$ ?

Q:

Calculating Distance Two straight roads diverge at an angle of $$65 ^ { \circ }$$ . Two cars leave the intersection at 2:00 P.M., one trav- eling at $$50 mi / h$$ and the other at $$30 mi / h$$ . How far apart are the cars at $$2 : 30$$ P.M.?

Q:

A puppy weighs $$0.85 lb$$ at birth, and each week he gains $$24 \%$$ in weight. Let $$a _ { n }$$ be his weight in pounds at the end of his $$n$$ th week of life.

(a) Find a formula for $$a _ { n }$$ .

(b) How much does the puppy weigh when he is $$6$$ weeks old?

(c) Is the sequence $$a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$$ arithmetic, geometric, or neither?

Q:

Ancestors A person has two parents, four grandparents, eight great-grandparents, and so on. What is the total number of a person's ancestors in $$15$$ generations?

Q:

Write the expression without using sigma notation, and then find the sum.

(a) $$\sum _ { n = 1 } ^ { 5 } ( 1 - n ^ { 2 } )$$

(b) $$\sum _ { n = 3 } ^ { 6 } ( - 1 ) ^ { n } 2 ^ { n - 2 }$$

Q:

23. Average Linh has obtained scores of $$82,75$$ , and $$71$$ on her midterm algebra exams. If the final exam counts twice as much as a midterm, what score must she make on her final exam to get an average score of $$80$$ ? (Assume that the maxi- mum possible score on each test is $$100$$ .)

Q:

The first term of a geometric sequence is $$25$$ , and the fourth term is $$\frac { 1 } { 5 }$$ .

(a) Find the common ratio $$r$$ and the fifth term.

(b) Find the partial sum of the first eight terms.

Q:

Mortgage Dr. Gupta is considering a $$30$$ -year mortgage at $$6 \%$$ interest. She can make payments of $$\ 3500$$ a month. What size loan can she afford?

Q:

Funding an Annuity How much money must be invested now at $$9 \%$$ per year, compounded semiannually, to fund an annuity of $$20$$ payments of $$\ 200$$ each, paid every $$6$$ months, the first payment being $$6$$ months from now?

Q:

Annual Savings Program Ursula opens a $$1$$ -year CD that yields $$5 \%$$ interest per year. She begins with a deposit of $$\ 5000$$ . At the end of each year when the CD matures, she re- invests at the same $$5 \%$$ interest rate, also adding $$10 \%$$ to the value of the CD from her other savings. (So for example, after the first year her CD has earned $$5 \%$$ of $$\ 5000$$ in interest, for a value of $$\ 5250$$ at maturity. She then adds $$10 \%$$ , or $$\ 525$$ , bringing the total value of her renewed CD to $$\ 5775$$ .)

(a) Find a recursive formula for the amount $$U _ { n }$$ in Ursula's $$CD$$ when she reinvests at the end of the $$n$$ th year.

(b) Find the first five terms of the sequence $$U _ { n }$$ . Does this appear to be a geometric sequence?

(c) Use the pattern you observed in (b) to find a formula for $$U _ { n }$$ .

(d) How much has she saved after $$10$$ years?

Q:

Nautical Miles Find the distance along an arc on the surface of the earth that subtends a central angle of $$1$$ minute ( $$1$$ minute $$= \frac { 1 } { 60 }$$ degree). This distance is called a nautical mile. (The radius of the earth is $$3960 mi$$ .)

Q:

A parabolic reflector for a car headlight forms a bowl shape that is $$6$$ in. wide at its opening and $$3$$ in. deep, as shown in the figure at the left. How far from the vertex should the filament of the bulb be placed if it is to be located at the focus?

Q:

35. Which of the points $$A ( 6,7 )$$ or $$B ( - 5,8 )$$ is closer to the origin?

Q:

Interest Rate John buys a stereo system for $$\ 640 .$$ He agrees to pay $$\ 32$$ a month for $$2$$ years. Assuming that interest is compounded monthly, what interest rate is he paying?

Q:

Velocity of a Ball If a ball is thrown straight up with a veloc- ity of $$40 ft / s$$ , its height (in $$ft$$ ) after $$t$$ seconds is given by $$y = 40 t - 16 t ^ { 2 }$$ . Find the instantaneous velocity when $$t = 2$$ .

Q:

Gas Station A gas station sells regular gas for $$\ 2.20$$ per gal- lon and premium gas for $$\ 3.00$$ a gallon. At the end of a busi- ness day $$280$$ gallons of gas had been sold, and receipts totaled $$\ 680$$ . How many gallons of each type of gas had been sold?

Q:

24. The maximum weight $$M$$ that can be supported by a beam is jointly proportional to its width $$w$$ and the square of its height $$h$$ and inversely proportional to its length $$L$$ .

(a) Write an equation that expresses this proportionality.

(b) Determine the constant of proportionality if a beam $$4$$ in. wide, $$6$$ in. high, and $$12 ft$$ long can support a weight of $$4800 lb$$ .

(c) If a 10-ft beam made of the same material is $$3$$ in. wide and $$10$$ in. high, what is the maximum weight it can support?

Q:

Work Done by a Winch A motorized winch is being used to pull a felled tree to a log- ging truck. The motor exerts a force of $$f ( x ) = 1500 + 10 x - \frac { 1 } { 2 } x ^ { 2 } lb$$ on the tree at the instant when the tree has moved $$x$$ ft. The tree must be moved a distance of $$40 ft$$ , from $$x = 0$$ to $$x = 40$$ . How much work is done by the winch in moving the tree?

Q:

21. Find an equation for the line with the given property.

(a) It passes through the point $$( 3 , - 6 )$$ and is parallel to the line $$3 x + y - 10 = 0$$ .

(b) It has $$x$$ -intercept $$6$$ and $$y$$ -intercept $$4$$ .

Q:

87. 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 $$1000 ft ^ { 3 }$$ of grain?

Q:

87. Global Warming Some scientists believe that the average surface temperature of the world has been rising steadily. The average surface temperature can be modeled by $$T = 0.02 t + 15.0$$

Q:

Deer Population The graph shows the deer population in a Pennsylvania county between $$2010$$ and $$2014$$ . Assume that the population grows exponentially.

(a) What was the deer population in $$2010$$ ?

(b) Find a function that models the deer population $$t$$ years after $$2010$$ .

(c) What is the projected deer population in $$2018 ?$$

Q:

Annuity Find the amount of an annuity that consists of $$16$$ quarterly payments of $$\ 300$$ each into an account that pays $$8 \%$$ interest per year, compounded quarterly.

Q:

14. A bottle of medicine is to be stored at a temperature between $$5 ^ { \circ } C$$ and $$10 ^ { \circ } C$$ . What range does this correspond to on the Fahrenheit scale? [Note: Fahrenheit $$( F )$$ and Celsius $$( C )$$ temperatures satisfy the relation $$C = \frac { 5 } { 9 } ( F - 32 ) . ]$$

Q:

101. Geometry A yellow square of side $$1$$ is divided into nine smaller squares, and the middle square is colored blue as shown in the figure. Each of the smaller yellow squares is in turn divided into nine squares, and each middle square is colored blue. If this process is continued indefinitely, what is the total area that is colored blue?

Q:

Area A rectangular box with a volume of $$60 ft ^ { 3 }$$ has a square base. Find a function that models its surface area $$S$$ in terms of the length $$x$$ of one side of its base.

Q:

Leaning Ladder A $$20$$ - $$ft$$ ladder is leaning against a building. If the base of the ladder is $$6 ft$$ from the base of the building, what is the angle of elevation of the ladder? How high does the ladder reach on the building?

Q:

31. Overtime Pay Helen earns $$\ 7.50$$ an hour at her job, but if she works more than $$35$$ hours in a week, she is paid $$1 \frac { 1 } { 2 }$$ times her regular salary for the overtime hours worked. One week her gross pay was $$\ 352.50$$ . How many overtime hours did she work that week?

Q:

The Leaning Tower of Pisa The bell tower of the cathedral in Pisa, Italy, leans $$5.6 ^ { \circ }$$ from the vertical. A tourist stands $$105 m$$ from its base, with the tower leaning directly toward her. She measures the angle of elevation to the top of the tower to be $$29.2 ^ { \circ } .$$ Find the length of the tower to the nearest meter.

Q:

Annuity What is the present value of an annuity that con- sists of $$20$$ semiannual payments of $$\ 1000$$ at an interest rate of $$9 \%$$ per year, compounded semiannually?

Q:

(a) Use the discriminant to determine whether the graph of the following equation is a parabola, an ellipse, or a hyperbola: $$5 x ^ { 2 } + 4 x y + 2 y ^ { 2 } = 18$$

Q:

71. Distance, Speed, and Time A pilot flew a jet from Montreal to Los Angeles, a distance of $$2500$$ mi. On the return trip, the average speed was $$20 \%$$ faster than the outbound speed. The round-trip took $$9 h 10$$ min. What was the speed from Mon- treal to Los Angeles?

Q:

Fan A ceiling fan with $$16$$ -in. blades rotates at $$45 rpm$$ .

(a) Find the angular speed of the fan in $$rad / min$$ .

(b) Find the linear speed of the tips of the blades in in./min.

Q:

Making Furniture A furniture manufacturer makes wooden tables and chairs. The pro- duction process involves two basic types of labor: carpentry and finishing. A table requires $$2 h$$ of carpentry and $$1 h$$ of finishing, and a chair requires $$3 h$$ of carpentry and $$\frac { 1 } { 2 } h$$ of finish- ing. The profit is $$\ 35$$ per table and $$\ 20$$ per chair. The manufacturer's employees can supply a maximum of $$108 h$$ of carpentry work and $$20 h$$ of finishing work per day. How many tables and chairs should be made each day to maximize profit?

Q:

Volume A rectangular box has a square base. Its height is half the width of the base. Find a function that models its volume $$V$$ in terms of its width $$w .$$

Q:

Let $$f ( x ) = \frac { \sqrt { x } } { x + 1 }$$

(a) Evaluate $$f ( 0 ) , f ( 2 )$$ , and $$f ( a + 2 )$$

(b) Find the domain of $$f$$ .

(c) What is the average rate of change of $$f$$ between $$x = 2$$ and $$x = 10$$ ?

Q:

Find the term containing $$x ^ { 3 }$$ in the binomial expansion of $$( 3 x - 2 ) ^ { 10 }$$

Q:

Radius Find a function that models the radius $$r$$ of a circle in terms of its area $$A$$ .

Q:

The first term of an arithmetic sequence is $$10$$ , and the tenth term is $$2 .$$

(a) Find the common difference and the $$100$$ th term of the sequence.

(b) Find the partial sum of the first ten terms.

Q:

Investments Clarisse invests $$\ 60,000$$ in money-market accounts at three different banks. Bank A pays $$2 \%$$ interest per year, bank B pays $$2.5 \%$$ , and bank C pays $$3 \%$$ . She decides to invest twice as much in bank B as in the other two banks. After $$1$$ year, Clarisse has earned $$\ 1575$$ in interest. How much did she invest in each bank?

Q:

39. Stopping Distance The stopping distance $$D$$ of a car after the brakes have been applied varies directly as the square of the speed $$s$$ . A certain car traveling at $$40 mi / h$$ can stop in $$150 ft$$ . What is the maximum speed it can be traveling if it needs to stop in $$200 ft$$ ?

Q:

Shipping Televisions An electronics discount chain has a sale on a certain brand of $$60$$ -in. high-definition television set. The chain has stores in Santa Monica and El Toro and warehouses in Long Beach and Pasadena. To satisfy rush orders, $$15$$ sets must be shipped from the warehouses to the Santa Monica store, and $$19$$ must be shipped to the El Toro store. The cost of shipping a set is $$\ 5$$ from Long Beach to Santa Monica, $$\ 6$$ from Long Beach to El Toro, $$\ 4$$ from Pasadena to Santa Monica, and $$\ 5.50$$ from Pasadena to El Toro.

Q:

Tchebycheff Polynomials

(a) Show that there is a polynomial $$P ( t )$$ of degree $$4$$ such that $$\cos 4 x = P ( \cos x )$$ (see Example $$2 )$$ .

(b) Show that there is a polynomial $$Q ( t )$$ of degree $$5$$ such that $$\cos 5 x = Q ( \cos x )$$ . [Note: In general, there is a polynomial $$P _ { n } ( t )$$ of degree $$n$$ such that cos $$n x = P _ { n } ( \cos x )$$ . These polynomials are called Tchebycheff polynonials, after the Russian mathematician P. L. Tchebycheff (1821-1894).]

Q:

45. Dimensions of a Lot A rectangular parcel of land is $$50 ft$$ wide. The length of a diagonal between opposite cor- ners is $$10 ft$$ more than the length of the parcel. What is the length of the parcel?

Q:

The line perpendicular to the $$x z$$ -plane that contains the point $$( 2 , - 1,5 )$$ .

Q:

Mortgage A couple secures a $$30$$ -year loan of $$\ 100,000$$ at $$9 \frac { 3 } { 4 } \%$$ per year, compounded monthly, to buy a house.

(a) What is the amount of their monthly payment?

(b) What total amount will they pay over the $$30$$ -year period?

(c) If, instead of taking the loan, the couple deposits the monthly payments in an account that pays $$9 \frac { 3 } { 4 } \%$$ interest per year, compounded monthly, how much will be in the account at the end of the $$30$$ -year period?

Q:

Wheelchair Ramp A local diner must build a wheelchair ramp to provide handicap access to the restaurant. Federal building codes require that a wheelchair ramp must have a maximum rise of $$1$$ in. for every horizontal distance of $$12$$ in.

(a) What is the maximum allowable slope for a wheelchair ramp? Assuming that the ramp has maximum rise, find a linear function $$H$$ that models the height of the ramp above the ground as a function of the horizontal distance $$x$$ .

(b) If the space available to build a ramp is $$150$$ in. wide, how high does the ramp reach?

Q:

Compound Interest Which of the given interest rates and compounding periods would provide the best investment?

(a) $$2 \frac { 1 } { 2 } \%$$ per year, compounded semiannually

(b) $$2 \frac { 1 } { 4 } \%$$ per year, compounded monthly

(c) $$2 \%$$ per year, compounded continuously

Q:

Minimizing Costs A rancher wants to build a rectangular pen with an area of $$100 m ^ { 2 }$$ .

(a) Find a function that models the length of fencing required.

(b) Find the pen dimensions that require the minimum amount of fencing.

Q:

Geometric Means If the numbers $$a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$$ form a geo- metric sequence, then $$a _ { 2 } , a _ { 3 } , \ldots , a _ { n - 1 }$$ are geometric means between $$a _ { 1 }$$ and $$a _ { n }$$ . Insert three geometric means between $$5$$ and $$80$$ .

Q:

Elevation of a Kite A man is lying on the beach, flying a kite. He holds the end of the kite string at ground level and estimates the angle of elevation of the kite to be $$50 ^ { \circ }$$ . If the string is $$450 ft$$ long, how high is the kite above the ground?

Q:

Mixing Coolant A truck radiator holds $$5$$ gal and is filled with water. A gallon of water is removed from the radiator and replaced with a gallon of antifreeze; then a gallon of the mixture is removed from the radiator and again replaced by a gallon of antifreeze. This process is repeated indefinitely. How much water remains in the tank after this process is repeated $$3$$ times? $$5$$ times? $$n$$ times?

Q:

Advertising The effectiveness of a television commercial depends on how many times a viewer watches it. After some experiments an advertising agency found that if the effective- ness $$E$$ is measured on a scale of $$0$$ to $$10$$ , then $$E ( n ) = \frac { 2 } { 3 } n - \frac { 1 } { 90 } n ^ { 2 }$$

Q:

Magnitude of Stars The magnitude $$M$$ of a star is a measure of how bright a star appears to the human eye. It is defined by $$M = - 2.5 \log ( \frac { B } { B _ { 0 } } )$$ where $$B$$ is the actual brightness of the star and $$B _ { 0 }$$ is a constant.

(a) Expand the right-hand side of the equation.

(b) Use part (a) to show that the brighter a star, the less its magnitude.

(c) Betelgeuse is about $$100$$ times brighter than Albiero. Use part (a) to show that Betelgeuse is $$5$$ magnitudes less bright than Albiero.

Q:

Work The force $$F = 4 i - 7 j$$ moves an object $$4 ft$$ along the $$x$$ -ax is in the positive direction. Find the work done if the unit of force is the pound.

Q:

61. Mixture Problem A bottle contains $$750 mL$$ of fruit punch with a concentration of $$50 \%$$ pure fruit juice. Jill drinks $$100 mL$$ of the punch and then refills the bottle with an equal amount of a cheaper brand of punch. If the concentration of juice in the bottle is now reduced to $$48 \%$$ , what was the con- centration in the punch that Jill added?

Q:

Earthquake Magnitude and Intensity

(a) Find the magnitude of an earthquake that has an intensity that is $$31.25$$ (that is, the amplitude of the seismograph reading is $$31.25 cm$$ ).

(b) An earthquake was measured to have a magnitude of $$4.8$$ on the Richter scale. Find the intensity of the earthquake.

Q:

Fish Farming A fish farmer has $$5000$$ catfish in his pond. The number of catfish increases by $$8 \%$$ per month, and the farmer harvests $$300$$ catfish per month.

(a) Show that the catfish population $$P _ { n }$$ after $$n$$ months is given recursively by $$P _ { 0 } = 5000$$ and $$P _ { n } = 1.08 P _ { n - 1 } - 300$$

Q:

Depth of Snowfall Snow began falling at noon on Sunday. The amount of snow on the ground at a certain location at time $$t$$ was given by the function $$h ( t ) = 11.60 t - 12.41 t ^ { 2 } + 6.20 t ^ { 3 }$$ $$- 1.58 t ^ { 4 } + 0.20 t ^ { 5 } - 0.01 t ^ { 6 }$$

Q:

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

Q:

11. Resultant Force Two tugboats are pulling a barge as shown in the figure. One pulls with a force of $$2.0 \times 10 ^ { 4 } lb$$ in the direction $$N 50 ^ { \circ } E$$ , and the other pulls with a force of $$3.4 \times 10 ^ { 4 } lb$$ in the direction $$S 75 ^ { \circ } E$$ .

(a) Find the resultant force on the barge as a vector.

(b) Find the magnitude and direction of the resultant force.

Q:

81. Mixtures The owner of a store sells raisins for $$\ 3.20$$ per pound and nuts for $$\ 2.40$$ per pound. He decides to mix the raisins and nuts and sell $$50$$ lb of the mixture for $$\ 2.72$$ per pound. What quantities of raisins and nuts should he use?

Q:

Saving How much money should be invested every quarter at $$10 \%$$ per year, compounded quarterly, to have $$\ 5000$$ in $$2$$ years?

Q:

Population of California The population of California was $$29.76$$ million in $$1990$$ and $$33.87$$ million in $$2000$$ . Assume that the population grows exponentially.

(a) Find a function that models the population $$t$$ years after $$1990$$ .

(b) Find the time required for the population to double.

(c) Use the function from part (a) to predict the population of California in the year $$2010$$ . Look up California's actual population in $$2010$$ , and compare.

Q:

Bonfire Temperature In the vicinity of a bonfire the temper- ature $$T ($$ in $${ } ^ { \circ } C )$$ at a distance of $$x$$ meters from the center of the fire is given by $$T ( x ) = \frac { 500,000 } { x ^ { 2 } + 400 }$$

Q:

119. Shifting the Coordinate Plane Suppose that each point in the coordinate plane is shifted $$3$$ units to the right and $$2$$ units upward.

(a) The point $$( 5,3 )$$ is shifted to what new point?

(b) The point $$( a , b )$$ is shifted to what new point?

Q:

Motion of a Building A strong gust of wind strikes a tall building, causing it to sway back and forth in damped har- monic motion. The frequency of the oscillation is $$0.5$$ cycle per second, and the damping constant is $$c = 0.9$$ . Find an equation that describes the motion of the building. (Assume that $$k = 1$$ , and take $$t = 0$$ to be the instant when the gust of wind strikes the building.)

Q:

Discover and Prove an Inequality Let $$F _ { n }$$ be the $$n$$ th term of the Fibonacci sequence. Find and prove an inequality relating $$n$$ and $$F _ { n }$$ for natural numbers $$n$$ .

Q:

Dead Reckoning A pilot flies in a straight path for $$1 h 30 min$$ . She then makes a course correction, heading $$10 ^ { \circ }$$ to the right of her original course, and flies $$2 h$$ in the new direction. If she maintains a constant speed of $$625 mi / h$$ , how far is she from her starting position?

Q:

Annual Percentage Yield Find the annual percentage yield for an investment that earns $$8 \%$$ per year, compounded monthly.

Q:

Orbit of the Earth The polar equation of an ellipse can be expressed in terms of its eccentricity $$e$$ and the length $$a$$ of its major axis.

(a) Show that the polar equation of an ellipse with directrix $$x = - d$$ can be written in the form $$r = \frac { a ( 1 - e ^ { 2 } ) } { 1 - e \cos \theta }$$ [Hint: Use the relation $$a ^ { 2 } = e ^ { 2 } d ^ { 2 } / ( 1 - e ^ { 2 } ) ^ { 2 }$$ given in the proof on page $$825 . ]$$

(b) Find an approximate polar equation for the elliptical orbit of the earth around the sun (at one focus) given that the eccentricity is about $$0.017$$ and the length of the major axis is about $$2.99 \times 10 ^ { 8 } km$$ .

Q:

53. Frequency of Vibration The frequency $$f$$ of vibration of a violin string is inversely proportional to its length $$L$$ . The constant of proportionality $$k$$ is positive and depends on the tension and density of the string.

(a) Write an equation that represents this variation.

(b) What effect does doubling the length of the string have on the frequency of its vibration?

Q:

133. Fish Population The fish population in a certain lake rises and falls according to the formula $$F = 1000 ( 30 + 17 t - t ^ { 2 } )$$

Q:

Terms of an Arithmetic Sequence The first term of an arithme- tic sequence is $$a = 7$$ , and the common difference is $$d = 3$$ . How many terms of this sequence must be added to obtain $$325 ?$$

Q:

Triangular Field A triangular field has sides of lengths $$22 ,$$ $$36 ,$$ and $$44 yd$$ . Find the largest angle.

Q:

Population of a City A city was incorporated in $$2004$$ with a population of $$35,000$$ . It is expected that the population will increase at a rate of $$2 \%$$ per year. The population $$n$$ years after $$2004$$ is given by $$P _ { n } = 35,000 ( 1.02 ) ^ { n }$$

Q:

Financing a Car A woman wants to borrow $$\ 12,000$$ to buy a car. She wants to repay the loan by monthly installments for $$4$$ years. If the interest rate on this loan is $$10 \frac { 1 } { 2 } \%$$ per year, compounded monthly, what is the amount of each payment?

Q:

123. Fencing a Garden A determined gardener has $$120 ft$$ of deer-resistant fence. She wants to enclose a rectangular veg- etable garden in her backyard, and she wants the area that is enclosed to be at least $$800 ft ^ { 2 }$$ . What range of values is pos- sible for the length of her garden?

Q:

Retirement Accounts Many college professors keep retirement savings with TIAA, the largest annuity program in the world. Interest on these accounts is compounded and cred- ited daily. Professor Brown has $$\ 275,000$$ on deposit with TIAA at the start of $$2015$$ and re- ceives $$3.65 \%$$ interest per year on his account.

(a) Find a recursive sequence that models the amount $$A _ { n }$$ in his account at the end of the $$n$$ th day of $$2015$$ .

(b) Find the first eight terms of the sequence $$A _ { n }$$ , rounded to the nearest cent.

(c) Find a formula for $$A _ { n }$$ .

Q:

Agriculture A farmer has $$1200$$ acres of land on which he grows corn, wheat, and soybeans. It costs $$\ 45$$ per acre to grow com, $$\ 60$$ to grow wheat, and $$\ 50$$ to grow soybeans. Because of market demand, the farmer will grow twice as many acres of wheat as of com. He has allocated $$\ 63,750$$ for the cost of growing his crops. How many acres of each crop should he plant?

Q:

Force A car is on a driveway that is inclined $$10 ^ { \circ }$$ to the hori- zontal. A force of $$490 lb$$ is required to keep the car from roll- ing down the driveway.

(a) Find the weight of the car.

(b) Find the force the car exerts against the driveway.

Q:

Buying Fruit A roadside fruit stand sells apples at $$75 \phi$$ a pound, peaches at $$90 \phi$$ a pound, and pears at $$60 \%$$ a pound. Muriel buys $$18 lb$$ of fruit at a total cost of $$\ 13.80$$ . Her peaches and pears together cost $$\ 1.80$$ more than her apples.

(a) Set up a linear system for the number of pounds of apples, peaches, and pears that she bought.

(b) Solve the system using Cramer's Rule.

Q:

Market Research A market analyst working for a small- appliance manufacturer finds that if the firm produces and sells $$x$$ blenders annually, the total profit (in dollars) is $$P ( x ) = 8 x + 0.3 x ^ { 2 } - 0.0013 x ^ { 3 } - 372$$ Graph the function $$P$$ in an appropriate viewing rectangle and use the graph to answer the following questions.

(a) When just a few blenders are manufactured, the firm loses money (profit is negative). (For example, $$P ( 10 ) = - 263.3$$ , so the firm loses $$\ 263.30$$ if it pro- duces and sells only $$10$$ blenders.) How many blenders must the firm produce to break even?

(b) Does profit increase indefinitely as more blenders are produced and sold? If not, what is the largest possible profit the firm could have?

Q:

Area of a Ripple A stone is dropped in a lake, creating a circular ripple that travels outward at a speed of $$60 cm / s$$ .

(a) Find a function $$g$$ that models the radius as a function of time.

(b) Find a function $$f$$ that models the area of the circle as a function of the radius.

(c) Find $$f \circ g .$$ What does this function represent?

Q:

69-70 Average Rate of Change A function $$f$$ is given. (a) Find the average rate of change of $$f$$ between $$x = 0$$ and $$x = 2$$ , and the average rate of change of $$f$$ between $$x = 15$$ and $$x = 50$$ . (b) Were the two average rates of change that you found in part (a) the same? (c) Is the function linear? If so, what is its rate of change?

Q:

Nutrition A researcher performs an experiment to test a hypothesis that involves the nutrients niacin and retinol. She feeds one group of laboratory rats a daily diet of precisely $$32$$ units of niacin and $$22,000$$ units of retinol. She uses two types of commercial pellet foods. Food A contains $$0.12$$ unit of niacin and $$100$$ units of retinol per gram. Food B contains $$0.20$$ unit of niacin and $$50$$ units of retinol per gram. How many grams of each food does she feed this group of rats each day?

Q:

Components of Force A man pushes a lawn mower with a force of $$30 lb$$ exerted at an angle of $$30 ^ { \circ }$$ to the ground. Find the horizontal and vertical components of the force.