Chapter 2

Express the statement as an equation. Use the given information to find the constant of proportionality. \(C\) is jointly proportional to \(I, w,\) and \(h .\) If \(I=w=h=2,\) then \(C=128 .\)

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ y=\sqrt{x} $$

A pair of points is graphed. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the mid-point of the segment that joins them. \((0,8),(6,16)\)

Find an equation of the line that satisfies the given conditions. Slope \(\frac{2}{3} ; \quad y\) -intercept 4

23-26 \(\mathbf{}\) Do the graphs intersect in the given viewing rectangle? If they do, how many points of intersection are there? $$ y=x^{3}-4 x, y=x+5 ; \quad[-4,4] \text { by }[-15,15] $$

Express the statement as an equation. Use the given information to find the constant of proportionality. \(H\) is jointly proportional to the squares of \(I\) and \(w .\) If \(I=2\) and \(w=\frac{1}{3},\) then \(H=36 .\)

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ x^{2}+y^{2}=9 $$

A pair of points is graphed. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the mid-point of the segment that joins them. \((-2,5),(10,0)\)

Find an equation of the line that satisfies the given conditions. \(x\) intercept \(1 ; \quad y\) intercept \(-3\)

Express the statement as an equation. Use the given information to find the constant of proportionality. \(s\) is inversely proportional to the square root of \(t\) . If \(s=100\) , then \(t=25 .\)

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ y=\sqrt{4-x^{2}} $$

A pair of points is graphed. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the mid-point of the segment that joins them. \((-3,-6),(4,18)\)

Find an equation of the line that satisfies the given conditions. \(x\) -intercept \(-8 ; \quad y\) intercept 6

Express the statement as an equation. Use the given information to find the constant of proportionality. \(M\) is jointly proportional to \(a, b,\) and \(c\) and inversely proportional to \(d\) If \(a\) and \(d\) have the same value and if \(b\) and \(c\) are both \(2,\) then \(M=128 .\)

Graph the circle \((y-1)^{2}+x^{2}=1\) by solving for \(y\) and graphing two equations as in Example \(3 .\)

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ y=-\sqrt{4-x^{2}} $$

A pair of points is graphed. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the mid-point of the segment that joins them. \((-1,-1),(9,9)\)

Find an equation of the line that satisfies the given conditions. Through \((4,5) ; \quad\) parallel to the \(x\) axis

Hooke's Law Hooke's Law states that the force needed to keep a spring stretched \(x\) units beyond its natural length is directly proportional to \(x\) . Here the constant of proportionality is called the spring constant. (a) Write Hooke's Law as an equation. (b) If a spring has a natural length of 10 \(\mathrm{cm}\) and a force of 40 \(\mathrm{N}\) is required to maintain the spring stretched to a length of \(15 \mathrm{cm},\) find the spring constant. (c) What force is needed to keep the spring stretched to a length of 14 \(\mathrm{cm} ?\)

Graph the equation \(4 x^{2}+2 y^{2}=1\) by solving for \(y\) and graphing two equations corresponding to the negative and positive square roots. (This graph is called an ellipse.)

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ y=|x| $$

A pair of points is graphed. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the mid-point of the segment that joins them. \((6,-2),(-1,3)\)

Find an equation of the line that satisfies the given conditions. Through \((4,5) ;\) parallel to the \(y\) axis

Law of the Pendulum The period of a pendulum (the time elapsed during one complete swing of the pendulum) varies directly with the square root of the length of the pendulum. (a) Express this relationship by writing an equation. (b) To double the period, how would we have to change the length \(R\).

Graph the equation \(y^{2}-9 x^{2}=1\) by solving for \(y\) and graphing the two equations corresponding to the positive and negative square roots. (This graph is called a hyperbola)

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ x=|y| $$

A pair of points is graphed. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the mid-point of the segment that joins them. \((-1,6),(-1,-3)\)

Find an equation of the line that satisfies the given conditions. Through \((1,-6) ;\) parallel to the line \(x+2 y=6\)

Printing Costs The cost \(C\) of printing a magazine is jointly proportional to the number of pages \(p\) in the magazine and the number of magazines printed \(m .\) (a) Write an equation that expresses this joint variation. (b) Find the constant of proportionality if the printing cost is \(\$ 60,000\) for 4000 copies of a 120 -page magazine. (c) How much would the printing cost be for 5000 copies of a 92 -page magazine?

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ y=4-|x| $$

A pair of points is graphed. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the mid-point of the segment that joins them. \((7,3),(11,6)\)

Find an equation of the line that satisfies the given conditions. y-intercept 6 : parallel to the line \(2 x+3 y+4=0\)

Boyle's Law The pressure \(P\) of a sample of gas is directly proportional to the temperature \(T\) and inversely proportional to the volume \(V\) (a) Write an equation that expresses this variation. (b) Find the constant of proportionality if 100 \(\mathrm{L}\) of gas exerts a pressure of 33.2 \(\mathrm{kPa}\) at a temperature of 400 \(\mathrm{K}\) (absolute temperature measured on the Kelvin scale). (c) If the temperature is increased to 500 \(\mathrm{K}\) and the volume is decreased to 80 \(\mathrm{L}\) , what is the pressure of the gas?

\(31-42\) . Solve the equation both algebraically and graphically. $$ \frac{1}{2} x-3=6+2 x $$

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ y=|4-x| $$

A pair of points is graphed. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the mid-point of the segment that joins them. \((2,13),(7,1)\)

Find an equation of the line that satisfies the given conditions. Through \((-1,2) ;\) parallel to the line \(x=5\)

Power from a Windmill The power \(P\) that can be obtained from a windmill is directly proportional to the cube of the wind speed \(s .\) (a) Write an equation that expresses this variation. (b) Find the constant of proportionality for a windmill that produces 96 watts of power when the wind is blowing at 20 \(\mathrm{mi} / \mathrm{h}\) . (c) How much power will this windmill produce if the wind speed increases to 30 \(\mathrm{mi} / \mathrm{h} ?\)

\(31-42\) . Solve the equation both algebraically and graphically. $$ \frac{2}{x}+\frac{1}{2 x}=7 $$

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ x=y^{3} $$

A pair of points is graphed. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the mid-point of the segment that joins them. \((3,4),(-3,-4)\)

Power Needed to Propel a Boat The power \(P\) (measured in horsepower, hp) needed to propel a boat is directly proportional to the cube of the speed \(s\) . An \(80-\) hp engine is needed to propel a certain boat at 10 knots. Find the power needed to drive the boat at 15 knots.

\(31-42\) . Solve the equation both algebraically and graphically. $$ \frac{4}{x+2}-\frac{6}{2 x}=\frac{5}{2 x+4} $$

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ y=x^{3}-1 $$

A pair of points is graphed. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the mid-point of the segment that joins them. \((5,0),(0,6)\)

Find an equation of the line that satisfies the given conditions. Through \((-1,-2) :\) perpendicular to the line \(2 x+5 y+8=0\)

Loudness of Sound The loudness \(L\) of a sound (measured in decibels, dB) is inversely proportional to the square of the distance \(d\) from the source of the sound. A person who is 10 \(\mathrm{ft}\) from a lawn mower experiences a sound level of 70 \(\mathrm{dB}\) . How loud is the lawn mower when the person is 100 \(\mathrm{ft}\) away?

\(31-42\) . Solve the equation both algebraically and graphically. $$ x^{2}-32=0 $$

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ y=x^{4} $$

Draw the rectangle with vertices \(A(1,3), B(5,3), C(1,-3),\) and \(D(5,-3)\) on a coordinate plane. Find the area of the rectangle.