Chapter 1

A nature conservancy group decides to construct a raised wooden walkway through a wetland area. To enclose the most interesting part of the wetlands, the walkway will have the shape of a right triangle with one leg 700 yd longer than the other and the hypotenuse 100 yd longer than the longer leg. Find the total length of the walkway.

Write each number in standard form a \(+b i\) $$\frac{-6-\sqrt{-24}}{2}$$

A student claims that the equation \(5 x=4 x\) is a contradiction, since dividing both sides by \(x\) leads to \(5=4,\) a false statement. Explain why the student is incorrect.

Solve each equation. $$\sqrt{3 x+7}=3 x+5$$

Solve each inequality. Write each solution set in interval notation. $$-3 \leq \frac{x-4}{-5}<4$$

Solve each equation by completing the square. $$x^{2}-4 x+3=0$$

Investing a Building Fund A church building fund has invested some money in two ways: part of the money at \(3 \%\) interest and four times as much at \(2.75 \% .\) Find the amount invested at each rate if the total annual income from interest is \(\$ 2800\).

Solve each inequality. Give the solution set using interval notation. $$|7-3 x| \leq 4$$

Problems involving the Pythagorean theorem have appeared in mathematics for thousands of years. This one is taken from the ancient Chinese work, Arithmetic in Nine Sections: There is a bamboo IO ft high, the upper end of which, being broken, reaches the ground 3 ft from the stem. Find the height of the break.

If \(k \neq 0,\) is the equation \(x+k=x\) a contradiction, a conditional equation, or an identity? Explain.

Solve each equation. $$\sqrt{4 x+13}=2 x-1$$

Solve each inequality. Write each solution set in interval notation. $$1 \leq \frac{4 x-5}{-2}<9$$

Solve each equation by completing the square. $$x^{2}-7 x+12=0$$

Solve each inequality. Give the solution set using interval notation. $$\left|\frac{2}{3} x+\frac{1}{2}\right| \leq \frac{1}{6}$$

A projectile is launched from ground level with an initial velocity of \(v_{0}\) feet per second. Neglecting air resistance, its height in feet \(t\) seconds after launch is given by $$s=-16 t^{2}+v_{0} t$$ Find the time(s) that the projectile will (a) reach a height of 80 \(\mathrm{ft}\). and (b) return to the ground for the given value of \(v_{0}\). Round answers to the nearest hundredth if necessary. $$v_{0}=96$$

Write each number in standard form a \(+b i\) $$\frac{10+\sqrt{-200}}{5}$$

Solve each quadratic inequality. Write each solution set in interval notation. $$x^{2}-x-6>0$$

Solve each equation. $$\sqrt{4 x+5}-6=2 x-11$$

Solve each equation by completing the square. $$2 x^{2}-x-28=0$$

Solve each inequality. Give the solution set using interval notation. $$\left|\frac{5}{3}-\frac{1}{2} x\right|>\frac{2}{9}$$

A projectile is launched from ground level with an initial velocity of \(v_{0}\) feet per second. Neglecting air resistance, its height in feet \(t\) seconds after launch is given by $$s=-16 t^{2}+v_{0} t$$ Find the time(s) that the projectile will (a) reach a height of 80 \(\mathrm{ft}\). and (b) return to the ground for the given value of \(v_{0}\). Round answers to the nearest hundredth if necessary. $$v_{0}=128$$

Write each number in standard form a \(+b i\) $$\frac{20+\sqrt{-8}}{2}$$

Solve each formula for the indicated variable. Assume that the denominator is not 0 if variables appear in the denominator. See Examples \(4(a)\) and \((b)\). \(I=P r t, \quad\) for \(P \quad\) (simple interest)

Solve each quadratic inequality. Write each solution set in interval notation. $$x^{2}-7 x+10>0$$

Solve each equation. $$\sqrt{6 x+7}-9=x-7$$

Solve each equation by completing the square. $$4 x^{2}-3 x-10=0$$

Warehouse Club Membership Membership warehouse clubs offer shoppers low prices, along with rewards of cash back on club purchases. If the yearly fee for a warehouse club membership is \(\$ 100\) and the reward rate is \(2 \%\) on club purchases for the year, then the linear equation $$ y=100-0.02 x $$ models the actual yearly cost of the membership \(y,\) in dollars. Here \(x\) represents the yearly amount of club purchases, also in dollars.(a) Determine the actual yearly cost of the membership if club purchases for the year are \(\$ 2400\) (b) What amount of club purchases would reduce the actual yearly cost of the membership to \(\$ 50 ?\) (c) How much would a member have to spend in yearly club purchases to reduce the yearly membership cost to \(\$ 0 ?\)

A projectile is launched from ground level with an initial velocity of \(v_{0}\) feet per second. Neglecting air resistance, its height in feet \(t\) seconds after launch is given by $$s=-16 t^{2}+v_{0} t$$ Find the time(s) that the projectile will (a) reach a height of 80 \(\mathrm{ft}\). and (b) return to the ground for the given value of \(v_{0}\). Round answers to the nearest hundredth if necessary. $$v_{0}=32$$

Write each number in standard form a \(+b i\) $$\frac{-3+\sqrt{-18}}{24}$$

Solve each formula for the indicated variable. Assume that the denominator is not 0 if variables appear in the denominator. See Examples \(4(a)\) and \((b)\). \(P=a+b+c, \quad\) for \(c \quad\) (perimeter of a triangle)

Solve each equation by completing the square. $$x^{2}-2 x-2=0$$

Explain why the equation \(|x|=\sqrt{x^{2}}\) has infinitely many solutions.

A projectile is launched from ground level with an initial velocity of \(v_{0}\) feet per second. Neglecting air resistance, its height in feet \(t\) seconds after launch is given by $$s=-16 t^{2}+v_{0} t$$ Find the time(s) that the projectile will (a) reach a height of 80 \(\mathrm{ft}\). and (b) return to the ground for the given value of \(v_{0}\). Round answers to the nearest hundredth if necessary. $$v_{0}=16$$

Write each number in standard form a \(+b i\) $$\frac{-5+\sqrt{-50}}{10}$$

Solve each quadratic inequality. Write each solution set in interval notation. $$3 x^{2}+x \leq 4$$

Solve each equation. $$\sqrt{2 x}-x+4=0$$

Solve each equation by completing the square. $$x^{2}-10 x+18=0$$

Solve each equation or inequality. $$|4 x+3|-2=-1$$

Indoor Air Pollution Formaldehyde is an indoor air pollutant formerly found in plywood, foam insulation, and carpeting. When concentrations in the air reach 33 micrograms per cubic foot ( \(\mu \mathrm{g} / \mathrm{ft}^{3}\) ), eye irritation can occur. One square foot of new plywood could emit \(140 ~ \mu g\) per hr. (Source: A. Hines, Indoor Air Quality \& Control. (a) A room has \(100 \mathrm{ft}^{2}\) of new plywood flooring. Find a linear equation \(F\) that computes the amount of formaldehyde, in micrograms, emitted in \(x\) hours. (b) The room contains \(800 \mathrm{ft}^{3}\) of air and has no ventilation. Determine how long it would take for concentrations to reach \(33 \mu \mathrm{g} / \mathrm{ft}^{3}\).

An astronaut on the moon throws a baseball upward. The astronaut is \(6 \mathrm{ft}, 6\) in. tall, and the initial velocity of the ball is \(30 \mathrm{ft}\) per sec. The height \(s\) of the ball in feet is given by the equation $$s=-2.7 t^{2}+30 t+6.5$$ where \(t\) is the number of seconds after the ball was thrown. (a) After how many seconds is the ball \(12 \mathrm{ft}\) above the moon's surface? Round to the nearest hundredth. (b) How many seconds will it take for the ball to return to the surface? Round to the nearest hundredth.

Find each sum or difference. Write the answer in standard form. $$(3+2 i)+(9-3 i)$$

Solve each quadratic inequality. Write each solution set in interval notation. $$-x^{2}-4 x-6 \leq-3$$

Solve each equation. $$\sqrt{x}-\sqrt{x-5}=1$$

Solve each equation by completing the square. $$2 x^{2}+x=10$$

Solve each equation or inequality. $$|8-3 x|-3=-2$$

Classroom Ventilation According to the American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (ASHRAE), a nonsmoking classroom should have a ventilation rate of \(15 \mathrm{ft}^{3}\) per min for each person in the room. (a) Write an equation that models the total ventilation \(V\) (in cubic feet per hour) necessary for a classroom with \(x\) students. (b) A common unit of ventilation is air change per hour (ach). 1 ach is equivalent to exchanging all of the air in a room every hour. If \(x\) students are in a classroom having volume \(15,000 \mathrm{ft}^{3},\) determine how many air exchanges per hour \((A)\) are necessary to keep the room properly ventilated. (c) Find the necessary number of ach ( \(A\) ) if the classroom has 40 students in it. (d) In areas like bars and lounges that allow smoking, the ventilation rate should be increased to \(50 \mathrm{ft}^{3}\) per min per person. Compared to classrooms, ventilation should be increased by what factor in heavy smoking areas?

Find each sum or difference. Write the answer in standard form. $$(4-i)+(8+5 i)$$

Solve each quadratic inequality. Write each solution set in interval notation. $$-x^{2}-6 x-16>-8$$

Solve each equation. $$\sqrt{x}-\sqrt{x-12}=2$$

Solve each equation by completing the square. $$3 x^{2}+2 x=5$$