Chapter 2

Find an equation of the line that satisfies the given conditions. Through \(\left(\frac{1}{2},-\frac{3}{3}\right);\) perpendicular to the line \(4 x-8 y=1\)

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 50 \(\mathrm{mi} / \mathrm{h}\) can stop in 240 \(\mathrm{ft}\) . What is the maximum speed it can be traveling if it needs to stop in 160 \(\mathrm{ft}\) ?

\(31-42\) . Solve the equation both algebraically and graphically. $$ x^{3}+16=0 $$

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ y=16-x^{4} $$

Draw the parallelogram with vertices \(A(1,2), B(5,2), C(3,6),\) and \(D(7,6)\) on a coordinate plane. Find the area of the parallelogram.

Find an equation of the line that satisfies the given conditions. Through \((1,7) ;\) parallel to the line passing through \((2,5)\) and \((-2,1)\)

A Jet of Water The power \(P\) of a jet of water is jointly proportional to the cross-sectional area \(A\) of the jet and to the cube of the velocity \(v\) . If the velocity is doubled and the cross- sectional area is halved, by what factor will the power increase?

\(31-42\) . Solve the equation both algebraically and graphically. $$ x^{2}+9=0 $$

\(37-40\) An equation and its graph are given. Find the \(x\) - and \(y\) -intercepts. $$ y=4 x-x^{2} $$

Plot the points \(A(1,0), B(5,0), C(4,3),\) and \(D(2,3)\) on a coordinate plane. Draw the segments \(A B, B C, C D,\) and \(D A\). What kind of quadrilateral is \(A B C D,\) and what is its area?

Find an equation of the line that satisfies the given conditions. Through \((-2,-11) ;\) perpendicular to the line passing through \((1,1)\) and \((5,-1)\)

Aerodynamic Lift The lift \(L\) on an airplane wing at take-off varies jointly as the square of the speed \(s\) of the plane and the area \(A\) of its wings. A plane with a wing area of 500 \(\mathrm{ft}^{2}\) traveling at 50 \(\mathrm{mi} / \mathrm{h}\) experiences a lift of 1700 \(\mathrm{Jb}\) . How much lift would a plane with a wing area of 600 \(\mathrm{ft}^{2}\) traveling at 40 \(\mathrm{mi} / \mathrm{h}\) experience?

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

\(37-40\) An equation and its graph are given. Find the \(x\) - and \(y\) -intercepts. $$ \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 $$

Plot the points \(P(5,1), Q(0,6),\) and \(R(-5,1)\) on a coordinate plane. Where must the point \(S\) be located so that the quadrilateral \(P Q R S\) is a square? Find the area of this square.

(a) Sketch the line with slope \(\frac{3}{2}\) that passes through the point \((-2,1)\) . (b) Find an equation for this line.

Drag Force on a Boat The drag force \(F\) on a boat is jointly proportional to the wetted surface area \(A\) on the hull and the square of the speed \(s\) of the boat. A boat experiences a drag force of 220 lb when traveling at 5 mi/h with a wetted surface area of 40 \(\mathrm{ft}^{2}\) . How fast must a boat be traveling if it has 28 \(\mathrm{ft}^{2}\) of wetted surface area and is experiencing a drag force of 175 \(\mathrm{lb} ?\)

\(31-42\) . Solve the equation both algebraically and graphically. $$ 16 x^{4}=625 $$

\(37-40\) An equation and its graph are given. Find the \(x\) - and \(y\) -intercepts. $$ x^{4}+y^{2}-x y=16 $$

Which of the points \(A(6,7)\) and \(B(-5,8)\) is closer to the origin?

(a) Sketch the line with slope \(-2\) that passes through the point \((4,-1)\) . (b) Find an equation for this line.

Skidding in a Curve A car is traveling on a curve that forms a circular arc. The force \(F\) needed to keep the car from skidding is jointly proportional to the weight \(w\) of the car and the square of its speed \(s,\) and is inversely proportional to the radius \(r\) of the curve. (a) Write an equation that expresses this variation. (b) A car weighing 1600 lb travels around a curve at 60 \(\mathrm{mi} / \mathrm{h}\) . The next car to round this curve weighs 2500 \(\mathrm{lb}\) and requires the same force as the first car to keep from skidding. How fast is the second car traveling?

\(31-42\) . Solve the equation both algebraically and graphically. $$ 2 x^{5}-243=0 $$

\(37-40\) An equation and its graph are given. Find the \(x\) - and \(y\) -intercepts. $$ x^{2}+y^{3}-x^{2} y^{2}=64 $$

Which of the points \(C(-6,3)\) and \(D(3,0)\) is closer to the point \(E(-2,1) ?\)

Use a graphing device to graph the given family of lines in the same viewing rectangle. What do the lines have in common? $$ y=-2 x+b \text { for } b=0, \pm 1, \pm 3, \pm 6 $$

Electrical Resistance The resistance \(R\) of a wire varies directly as its length \(L\) and inversely as the square of its diameter \(d\). (a) Write an equation that expresses this joint variation. (b) Find the constant of proportionality if a wire 1.2 \(\mathrm{m}\) long and 0.005 \(\mathrm{m}\) in diameter has a resistance of 140 ohms. (c) Find the resistance of a wire made of the same material that is 3 \(\mathrm{m}\) long and has a diameter of \(0.008 \mathrm{m} .\)

\(31-42\) . Solve the equation both algebraically and graphically. $$ (x-5)^{4}-80=0 $$

\(41-48\) Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$ y=x-3 $$

Which of the points \(P(3,1)\) and \(Q(-1,3)\) is closer to the point \(R(-1,-1) ?\)

Use a graphing device to graph the given family of lines in the same viewing rectangle. What do the lines have in common? $$ y=m x-3 \text { for } m=0, \pm 0.25, \pm 0.75, \pm 1.5 $$

Kepler's Third Law Kepler's Third Law of planetary motion states that the square of the period \(T\) of a planet (the time it takes for the planet to make a complete revolution about the sun) is directly proportional to the cube of its average distance \(d\) from the sun. (a) Express Kepler's Third Law as an equation. (b) Find the constant of proportionality by using the fact that for our planet the period is about 365 days and the average distance is about 93 million miles. (c) The planet Neptune is about \(2.79 \times 10^{9}\) mi from the sun. Find the period of Neptune.

\(31-42\) . Solve the equation both algebraically and graphically. $$ 6(x+2)^{5}=64 $$

\(41-48\) Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$ y=x^{2}-5 x+6 $$

(a) Show that the points \((7,3)\) and \((3,7)\) are the same distance from the origin. (b) Show that the points \((a, b)\) and \((b, a)\) are the same distance from the origin.

Use a graphing device to graph the given family of lines in the same viewing rectangle. What do the lines have in common? $$ y=m(x-3) \quad \text { for } m=0, \pm 0.25, \pm 0.75, \pm 1.5 $$

Radiation Energy The total radiation energy \(E\) emitted by a heated surface per unit area varies as the fourth power of its absolute temperature \(T\) . The temperature is 6000 \(\mathrm{K}\) at the surface of the sun and 300 \(\mathrm{K}\) at the surface of the earth. (a) How many times more radiation energy per unit area is produced by the sun than by the earth? (b) The radius of the earth is 3960 mi and the radius of the sun is \(435,000\) mi. How many times more total radiation does the sun emit than the earth?

\(43-50\) . Solve the equation graphically in the given interval. State each answer rounded to two decimals. $$ x^{2}-7 x+12=0 ; \quad[0,6] $$

\(41-48\) Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$ y=x^{2}-9 $$

Show that the triangle with vertices \(A(0,2), B(-3,-1),\) and \(C(-4,3)\) is isosceles.

Use a graphing device to graph the given family of lines in the same viewing rectangle. What do the lines have in common? $$ y=2+m(x+3) \text { for } m=0, \pm 0.5, \pm 1, \pm 2, \pm 6 $$

Value of a Lot The value of a building lot on Galiano Island is jointly proportional to its area and the quantity of water produced by a well on the property. A 200 \(\mathrm{ft}\) by 300 \(\mathrm{ft}\) lot has a well producing 10 gallons of water per minute, and is valued at \(\$ 48,000\) . What is the value of a 400 \(\mathrm{ft}\) by 400 \(\mathrm{ft}\) lot if the well on the lot produces 4 gallons of water per minute?

\(43-50\) . Solve the equation graphically in the given interval. State each answer rounded to two decimals. $$ x^{2}-0.75 x+0.125=0 ;[-2,2] $$

\(41-48\) Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$ y-2 x y+2 x=1 $$

Find the slope and y-intercept of the line, and draw its graph. $$ x+y=3 $$

Growing Cabbages In the short growing season of the Canadian arctic territory of Nunavut, some gardeners find it possible to grow gigantic cabbages in the midnight sun. Assume that the final size of a cabbage is proportional to the amount of nutrients it receives and inversely proportional to the number of other cabbages surrounding it. A cabbage that received 20 oz of nutrients and had 12 other cabbages around it grew to 30 lb. What size would it grow to if it received 10 oz of nutrients and had only 5 cabbage "neighbors"?

\(43-50\) . Solve the equation graphically in the given interval. State each answer rounded to two decimals. $$ x^{3}-6 x^{2}+11 x-6=0 ; \quad[-1,4] $$

\(41-48\) Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$ x^{2}+y^{2}=4 $$

Find the slope and y-intercept of the line, and draw its graph. $$ 3 x-2 y=12 $$

Heat of a Campfire The heat experienced by a hiker at a campfire is proportional to the amount of wood on the fire and inversely proportional to the cube of his distance from the fire. If he is 20 \(\mathrm{ft}\) from the fire and someone doubles the amount of wood burning, how far from the fire would he have to be so that he feels the same heat as before?