Chapter 6

Approximate, to the nearest thousandth. any solutions to the nonlinear system of equations graphically. $$\begin{aligned}x^{4}-3 x^{3} &=y \\\\\log x^{2}-y &=0\end{aligned}$$

Approximate, to the nearest thousandth. any solutions to the nonlinear system of equations graphically. $$\begin{aligned}&e^{2 x}+y=4\\\&\ln x-2 y=0\end{aligned}$$

In \(2000,\) the combined population of Minneapolis/St. Paul, Minnesota, was \(670,000\). The population of Minneapolis was \(96,000\) greater than the population of St. Paul. (a) Write a system of equations whose solution gives the population of each city in thousands. (b) Solve the system of equations. (c) Is your system consistent or inconsistent? If it is consistent, state whether the equations are dependent or independent.

In 2006 , the commercial sector used 3.74 quadrillion (10") Btu of energy from natural gas and petroleum. It used 3.02 quadrillion Btu more natural gas than petroleum. (a) Write a system of equations whose solution gives the consumption of natural gas and petroleum (in quadrillion Bta). (b) Solve the system of equations. (c) Is your system consistent or inconsistent? If it is consistent, state whether the equations are dependent or independent.

From 2001 to 2005 the average number of hours that a user spent on the Internet each week increased by \(110 \% .\) This percent increase amounted to 11 hours. Find the average number of hours that a user spent on the Internet each week in 2001 and 2005.

Find the radius and height of a cylindrical container with a volume of 50 cubic inches and a lateral surface area of 65 square inches.

Determine if it is possible to construct a cylindrical container, including the top and bottom, with a volume of 38 cubic inches and a surface area of 38 square inches.

A box has an open top, rectangular sides, and a square base. Its volume is 576 cubic inches, and its outside surface area is 336 square inches. Find the dimensions of the box.

A box has rectangular sides, and its rectangular top and base are twice as long as they are wide. Its volume is 588 cubic inches, and its outside surface area is 448 square inches. Find its dimensions.

The total number of robberies in 2000 and 2001 was \(831,000 .\) From 2000 to 2001 the number of robberies declined by \(15,000 .\) (a) Write a system of equations whose solution represents the number of robberies committed in each of these years. (b) Solve the system symbolically. (c) Solve the system graphically.

During the first four months of \(2004,\) a total of 6767 people were vaccinated for smallpox in Florida and Texas. There were 273 more people vaccinated in Florida than in Texas. (Source: CDC.) (a) Write a system of equations whose solution represents the number of vaccinations given in each state. (b) Solve the system symbolically. (c) Solve the system graphically.

A student takes out two loans totaling \(\$ 3000\) to help pay for college expenses. One loan is at \(8 \%\) interest, and the other is at \(10 \% .\) Interest for both loans is compounded annually. (a) If the first-year interest is \(\$ 264\), write a system of equations whose solution is the amount of each loan (b) Find the amount of each loan.

A student invests \(\$ 5000\) at two annual interest rates, \(5 \%\) and \(7 \% .\) After 1 year the student receives a total of \(\$ 325\) in interest. How much did the student invest at each interest rate?

Suppose a rectangular pen for a pet is to be made using 40 feet of fence. Let \(l\) represent its length and \(w\) its width, with \(l \geq w\) (a) Find \(l\) and \(w\) if the area is 91 square feet. (b) Write a formula for the area \(A\) in terms of \(w\) (c) What is the maximum area possible for the pen? Interpret this result.

A tugboat can pull a barge 60 miles upstream in 15 hours. The same tugboat and barge can make the return trip downstream in 6 hours. Determine the speed of the current in the river.

A jet airliner travels 1680 miles in 3 hours with a tail wind. The return trip, into the wind, takes 3.5 hours. Find both the speed of the jet with no wind and the wind speed.

A plane flies 1500 miles against the wind in 3 hours and 45 minutes. The return trip with the wind takes 3 hours. Assume that the wind speed stays constant. Find the speed of the wind and the speed of the airplane with no wind.

American battlefield deaths in World Wars I and II totaled about \(345,000\). There were about 5.5 times as many deaths in World War II as World War I. Find the number of American battlefield deaths in each war. Round your answers to the nearest whole number.

The total number of cigarettes sold overseas in 1990 and 2000 was 10.9 trillion cigarettes. The number sold in 2000 was 100 billion more than the number sold in \(1990 .\) How many cigarettes were sold each year?

The surface area of the skin covering the human body is a function of more than one variable. A taller person tends to have a larger surface area, as does a heavier person. Both height and weight influence the surface area of a person's body. A formula to determine the surface area of a person's body in square meters is given by \(S(w, h)=0.007184 w^{0.425} h^{0.725},\) where \(w\) is weight in kilograms and \(h\) is height in centimeters. Use \(S\) to estimate the surface area of a person who is 65 inches (165.1 centimeters) tall and weighs 154 pounds (70 kilograms).

Approximate the constant of variation to the nearest hundredth. The variable \(z\) varies jointly as the second power of \(x\) and the third power of \(y\). When \(x=2\) and \(y=2.5, z=31.9\).

Approximate the constant of variation to the nearest hundredth. The variable \(z\) varies jointly as the 1.5 power of \(x\) and the 2.1 power of \(y .\) When \(x=4\) and \(y=3.5, z=397\).

The variable \(z\) varies jointly as the square root of \(x\) and the cube root of \(y .\) If \(z=10.8\) when \(x=4\) and \(y=8\) find \(z\) when \(x=16\) and \(y=27\).

The variable \(z\) varies jointly as the third powers of \(x\) and y. If \(z=2160\) when \(x=3\) and \(y=4,\) find \(z\) when \(x=2\) and \(y=5\).

The electrical power generated by a windmill varies jointly as the square of the diameter of the area swept out by the blades and the cube of the wind velocity. If a windmill with an 8 -foot diameter and a 10 -mile-per-hour wind generates 2405 watts, how much power would be generated if the blades swept out an area 6 feet in diameter and the wind was 20 miles per hour?

The strength of a rectangular beam varies jointly as its width and the square of its thickness. If a beam 5.5 inches wide and 2.5 inches thick supports 600 pounds, how much can a similar beam that is 4 inches wide and 1.5 inches thick support?

The cost of carpet for a rectangular room varies jointly as its width and length. If a room 10 feet wide and 12 feet long costs \(\$ 1560\) to carpet, find the cost to carpet a room that is 11 feet by 23 feet. Interpret the constant of variation.

Give an example of a quantity occurring in everyday life that can be computed by a function of more than one input. Identify the inputs and the output.

Give an example of a system of linear equations with two variables. Explain how to solve the system graphically and symbolically.