Chapter 9

The solutions have been started for you. The first few exercises are each modeled by a system of two linear equations in two variables. Three times one number minus a second is 8 , and the sum of the numbers is 12 . Find the numbers. Start the solution: 1\. UNDERSTAND the problem. Since we are looking for two numbers, let \(x=\) one number \(y=\) second number 2\. TRANSLATE. Since we have assigned two variables, we will translate the facts into two equations. (Fill in the blanks.) 3\. SOLVE the system and 4\. INTERPRET the results.

Perform each indicated operation. $$ 5^{2}-11+3(-5) $$

Write each statement as an equation. Use \(k\) as the constant of variation. \(a\) varies jointly as \(b\) and \(c\).

The United States has the world's only "large deck" aircraft carriers, which can hold up to 72 aircraft. The Enterprise class carrier is longest in length while the Nimitz class carrier is the second longest. The total length of these two carriers is 2193 feet while the difference of their lengths is only 9 feet. (Source: USA Today, May 2001) a. Find the length of each class carrier. b. If a football field has a length of 100 yards, determine the length of the Enterprise class carrier in terms of number of football fields.

Perform each indicated operation. $$ 8^{2}+(-13)-4(-2) $$

For each statement, find the constant of variation and the variation equation. \(y\) varies directly as the cube of \(x ; y=9\) when \(x=3\)

The rate of growth of participation of women in sports has been increasing since Title IX was enacted in the 1970 s. In 2008 , the number of women participating in swimming was 1.1 million less than twice the number participating in running. If the total number of women participating in these two sports was 50.5 million, find the number of participants in each sport. (Source: Sporting Goods Association of America, 2009 report)

Perform each indicated operation. $$ (-12)^{2}+(-1)(2)-6 $$

The most popular amusement park in the world (according to annual attendance) is Tokyo Disneyland, whose yearly attendance in thousands can be approximated by the equation \(y=1201 x+16,507\) where \(x\) is the number of years after 2000 . In second place is Walt Disney World's Magic Kingdom, whose yearly attendance, in thousands, can be approximated by \(y=-616 x+15,400\). Find the last year when attendance in Magic Kingdom was greater than attendance in Tokyo Disneyland. (Source: Amusement Business)

For each statement, find the constant of variation and the variation equation. \(y\) varies directly as the cube of \(x ; y=32\) when \(x=4\)

Describe the solution of the system: \(\left\\{\begin{array}{l}y \leq 3 \\ y \geq 3\end{array}\right.\)

For the system \(\left\\{\begin{aligned} 2 x-3 y &=8 \\ x+5 y &=-3 \end{aligned}\right.\), explain what is wrong with writing the corresponding matrix as \(\left[\begin{array}{ccc}2 & 3 & 8 \\ 0 & 5 & -3\end{array}\right]\).

Gerry Gundersen mixes different solutions with concentrations of \(25 \%, 40 \%,\) and \(50 \%\) to get 200 liters of a \(32 \%\) solution. If he uses twice as much of the \(25 \%\) solution as of the \(40 \%\) solution, find how many liters of each kind he uses.

Describe the solution of the system: \(\left\\{\begin{array}{l}x \leq 5 \\ x \leq 3\end{array}\right.\)

For the system \(\left\\{\begin{array}{l}5 x+2 y=0 \\ -y \quad=2\end{array},\right.\) explain what is wrong with writing the corresponding matrix as \(\left[\begin{array}{rrr}5 & 2 & 0 \\ -1 & 0 & 2\end{array}\right]\).

For each statement, find the constant of variation and the variation equation. \(y\) varies directly as the square root of \(x ; y=2.1\) when \(x=4\) \(x=9\)

Gerry Gundersen mixes different solutions with concentrations of \(25 \%, 40 \%,\) and \(50 \%\) to get 200 liters of a \(32 \%\) solution. If he uses twice as much of the \(25 \%\) solution as of the \(40 \%\) solution, find how many liters of each kind he uses.

Explain how to decide which region to shade to show the solution region of the following system. $$ \left\\{\begin{array}{l} x \geq 3 \\ y \geq-2 \end{array}\right. $$

For each statement, find the constant of variation and the variation equation. \(y\) varies inversely as the square of \(x ; y=0.052\) when \(x=5\)

The measure of the largest angle of a triangle is \(90^{\circ}\) more than the measure of the smallest angle, and the measure of the remaining angle is \(30^{\circ}\) more than the measure of the smallest angle. Find the measure of each angle.

Tony Noellert budgets his time at work today. Part of the day he can write bills; the rest of the day he can use to write purchase orders. The total time available is at most 8 hours. Less than 3 hours is to be spent writing bills. a. Write a system of inequalities to describe the situation. (Let \(x=\) hours available for writing bills and \(y=\) hours available for writing purchase orders.) b. Graph the solutions of the system.

For each statement, find the constant of variation and the variation equation. \(y\) varies inversely as the square of \(x ; y=0.011\) when \(x=10\)

The sum of three numbers is 40 . The first number is five more than the second number. It is also twice the third. Find the numbers.

For each statement, find the constant of variation and the variation equation. \(y\) varies jointly as \(x\) and the cube of \(z ; y=120\) when \(x=5\) and \(z=2\)

The sum of the digits of a three-digit number is 15 . The tens-place digit is twice the hundreds-place digit, and the ones-place digit is 1 less than the hundredsplace digit. Find the three-digit number.

During the \(2008-2009\) regular NBA season, the top-scoring player was Dwyane Wade of the Miami Heat. Wade scored a total of 2386 points during the regular season. The number of free throws (each worth one point) he made was 26 less than seven times the number of three-point field goals he made. The number of two-point field goals that Wade made was 176 more than the number of free throws he made. How many free throws, two-point field goals, and three-point field goals did Dwyane Wade make during the \(2008-2009\) NBA season? (Source: National Basketball Association)

The maximum weight that a rectangular beam can support varies jointly as its width and the square of its height and inversely as its length. If a beam \(\frac{1}{2}\) foot wide, \(\frac{1}{3}\) foot high, and 10 feet long can support 12 tons, find how much a similar beam can support if the beam is \(\frac{2}{3}\) foot wide, \(\frac{1}{2}\) foot high, and 16 feet long.

For \(2009,\) the WNBA's top scorer was Cappie Poindexter of the Phoenix Mercury. She scored a total of 648 points during the regular season. The number of two-point field goals that Poindexter made was 22 fewer than five times the number of three-point field goals she made. The number of free throws (each worth one point) she made was 60 fewer than the number of two- point field goals she made. Find how many field goals, three-point field goals, and free throws Cappie Poindexter made during the 2009 regular season. (Source: Women's National Basketball Association)

The number of cars manufactured on an assembly line at a General Motors plant varies jointly as the number of workers and the time they work. If 200 workers can produce 60 cars in 2 hours, find how many cars 240 workers should be able to make in 3 hours.

The volume of a cone varies jointly as its height and the square of its radius. If the volume of a cone is \(32 \pi\) cubic inches when the radius is 4 inches and the height is 6 inches, find the volume of a cone when the radius is 3 inches and the height is 5 inches.

When a wind blows perpendicularly against a flat surface, its force is jointly proportional to the surface area and the speed of the wind. A sail whose surface area is 12 square feet experiences a 20 -pound force when the wind speed is 10 miles per hour. Find the force on an 8-square-foot sail if the wind speed is 12 miles per hour.

Multiply both sides of equation (1) by 2, and add the resulting equation to equation (2). $$ \begin{array}{l} 3 x-y+z=2 \\ -x+2 y+3 z=6 \end{array} $$

The intensity of light (in foot-candles) varies inversely as the square of \(x\), the distance in feet from the light source. The intensity of light 2 feet from the source is 80 foot-candles. How far away is the source if the intensity of light is 5 foot-candles?

Multiply both sides of equation (1) by 2, and add the resulting equation to equation (2). $$ \begin{aligned} &2 x+y+3 z=7\\\ &-4 x+y+2 z=4 \end{aligned} $$

The horsepower that can be safely transmitted to a shaft varies jointly as the shaft's angular speed of rotation (in revolutions per minute) and the cube of its diameter. A 2 -inch shaft making 120 revolutions per minute safely transmits 40 horsepower. Find how much horsepower can be safely transmitted by a 3 -inch shaft making 80 revolutions per minute.

Multiply both sides of equation (1) by -3 , and add the resulting equation to equation (2). $$ \begin{array}{r} x+2 y-z=0 \\ 3 x+y-z=2 \end{array} $$

Write an equation to describe each variation. Use \(k\) for the constant of proportionality. \(y\) varies directly as \(x\)

Multiply both sides of equation (1) by -3 , and add the resulting equation to equation (2). $$ \begin{aligned} 2 x-3 y+2 z &=5 \\ x-9 y+z &=-1 \end{aligned} $$

Write an equation to describe each variation. Use \(k\) for the constant of proportionality. \(p\) varies directly as \(q\)

Write a single linear equation in three variables that has (-1,2,-4) as a solution. (There are many possibilities.) Explain the process you used to write an equation.

Write an equation to describe each variation. Use \(k\) for the constant of proportionality. \(a\) varies inversely as \(b\).

When solving a system of three equations in three unknowns, explain how to determine that a system has no solution.

Write an equation to describe each variation. Use \(k\) for the constant of proportionality. \(y\) varies inversely as \(x\)

Write a system of linear equations in three variables that has the solution (-1,2,-4) . (There are many possibilities.) Explain the process you used to write your system.

Write an equation to describe each variation. Use \(k\) for the constant of proportionality. \(y\) varies jointly as \(x\) and \(z\)

Write a system of three linear equations in three variables that has (2,1,5) as a solution. (There are many possibilities.) Explain the process you used to write an equation.

Write an equation to describe each variation. Use \(k\) for the constant of proportionality. \(y\) varies jointly as \(q, r,\) and \(t\)

The fraction \(\frac{1}{24}\) can be written as the following sum: \(\frac{1}{24}=\frac{x}{8}+\frac{y}{4}+\frac{z}{3}\) where the numbers \(x, y,\) and \(z\) are solutions of \(\left\\{\begin{aligned} x+y+z=& 1 \\ 2 x-y+z=& 0 \\\\-x+2 y+2 z=&-1 \end{aligned}\right.\) Solve the system and see that the sum of the fractions is \(\frac{1}{24}\).

Write an equation to describe each variation. Use \(k\) for the constant of proportionality. \(y\) varies inversely as \(x^{3}\)

The fraction \(\frac{1}{18}\) can be written as the following sum: \(\frac{1}{18}=\frac{x}{2}+\frac{y}{3}+\frac{z}{9}\) where the numbers \(x, y,\) and \(z\) are solutions of \(\left\\{\begin{aligned} x+3 y+z &=-3 \\\\-x+y+2 z &=-14 \\ 3 x+2 y-z &=12 \end{aligned}\right.\) Solve the system and see that the sum of the fractions is \(\frac{1}{18}\).