Chapter 2

The population density \(D\) (in people/ \(\mathrm{mi}^{2}\) ) in a large city is related to the distance \(x\) from the center of the city by \(D=5000 x /\left(x^{2}+36\right)\). In what areas of the city does the population density exceed 400 people/ \(\mathrm{mi}^{2}\) ?

Exer. 1-50: Solve the equation. $$ \left(\frac{x}{x-2}\right)^{2}-\frac{2 x}{x-2}-15=0 $$

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ |x| \leq 7 $$

Exer. \(39-56:\) Find the solutions of the equation. $$ x^{3}-27=0 $$

Exer. 45-48: Use the quadratic formula to factor the expressions. $$ 15 x^{2}+34 x-16 $$

Show that the equation is an identity. $$\frac{3 x^{2}+8}{x}=\frac{8}{x}+3 x$$

After an astronaut is launched into space, the astronaut's weight decreases until a state of weightlessness is achieved. The weight of a 125 -pound astronaut at an altitude of \(x\) kilometers above sea level is given by $$ W=125\left(\frac{6400}{6400+x}\right)^{2} $$ At what altitudes is the astronaut's weight less than 5 pounds?

Exer. 1-50: Solve the equation. $$ \sqrt[3]{x}=2 \sqrt[4]{x} $$

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ |x| \geq 5 $$

Exer. \(39-56:\) Find the solutions of the equation. $$ 27 x^{3}=(x+5)^{3} $$

Exer. 49-50: Use the quadratic formula to solve the equation for (a) \(x\) in terms of \(y\) and (b) \(y\) in terms of \(x\). $$ 4 x^{2}-4 x y+1-y^{2}=0 $$

Show that the equation is an identity. $$\frac{49 x^{2}-25}{7 x-5}=7 x+5$$

The Lorentz contraction formula in relativity theory relates the length \(L\) of an object moving at a velocity of \(v \mathrm{mi} / \mathrm{sec}\) with respect to an observer to its length \(L_{0}\) at rest. If \(c\) is the speed of light, then $$ L^{2}=L_{0}^{2}\left(1-\frac{v^{2}}{c^{2}}\right) $$ For what velocities will \(L\) be less than \(\frac{1}{2} L_{0}\) ? State the answer in terms of \(c\).

Exer. 1-50: Solve the equation. $$ \sqrt{x+3}=\sqrt[4]{2 x+6} $$

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ |-x|>2 $$

Exer. \(39-56:\) Find the solutions of the equation. $$ 16 x^{4}=(x-4)^{4} $$

Exer. 49-50: Use the quadratic formula to solve the equation for (a) \(x\) in terms of \(y\) and (b) \(y\) in terms of \(x\). $$ 2 x^{2}-x y=3 y^{2}+1 $$

For what value of \(c\) is the number \(a\) a solution of the equation? $$4 x+1+2 c=5 c-3 x+6 ; \quad a=-2$$

In the design of certain small turbo-prop aircraft, the landing speed \(V\) (in \(\mathrm{ft} / \mathrm{sec})\) is determined by the formula \(W=0.00334 V^{2} S\), where \(W\) is the gross weight (in pounds) of the aircraft and \(S\) is the surface area (in \(\mathrm{ft}^{2}\) ) of the wings. If the gross weight of the aircraft is between 7500 pounds and 10,000 pounds and \(S=210 \mathrm{ft}^{2}\), determine the range of the landing speeds in miles per hour.

Exer. 51-52: Find the real solutions of the equation. (a) \(x^{5 / 3}=32\) (b) \(x^{4 / 3}=16\) (c) \(x^{2 / 3}=-36\) (d) \(x^{3 / 4}=125\) (e) \(x^{3 / 2}=-27\)

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ |x+3|<0.01 $$

Exer. \(39-56:\) Find the solutions of the equation. $$ x^{4}=256 $$

Exer. 51-54: Solve for the specified variable. $$ K=\frac{1}{2} m v^{2} \text { for } v $$ (kinetic energy)

For what value of \(c\) is the number \(a\) a solution of the equation? $$3 x-2+6 c=2 c-5 x+1 ; \quad a=4$$

Exer. 51-52: Find the real solutions of the equation. (a) \(x^{3 / 5}=-27\) (b) \(x^{2 / 3}=25\) (c) \(x^{4 / 3}=-49\) (d) \(x^{3 / 2}=27\) (e) \(x^{3 / 4}=-8\)

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ |x-4| \leq 0.03 $$

Exer. \(39-56:\) Find the solutions of the equation. $$ x^{4}=81 $$

Exer. 51-54: Solve for the specified variable. \(F=g \frac{m M}{d^{2}}\) for \(d\) (Newton's law of gravitation)

Exer. 53-56: Solve for the specified variable. $$ T=2 \pi \sqrt{\frac{l}{g}} \text { for } l \quad \text { (period of a pendulum) } $$

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ |x+2|+0.1 \geq 0.2 $$

Exer. \(39-56:\) Find the solutions of the equation. $$ 4 x^{4}+25 x^{2}+36=0 $$

Exer. 51-54: Solve for the specified variable. $$ A=2 \pi r(r+h) \text { for } r $$ (surface area of a closed cylinder)

Determine whether the two equations are equivalent. (a) \(\frac{8 x}{x-7}=\frac{72}{x-7}, \quad x=9\) (b) \(\frac{8 x}{x-7}=\frac{56}{x-7}\) \(x=7\)

Exer. 53-56: Solve for the specified variable. $$ d=\frac{1}{2} \sqrt{4 R^{2}-C^{2}} \text { for } C \quad \text { (segments of circles) } $$

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ |x-3|-0.3>0.1 $$

Exer. \(39-56:\) Find the solutions of the equation. $$ 27 x^{4}+21 x^{2}+4=0 $$

Exer. 51-54: Solve for the specified variable. $$ s=\frac{1}{2} g t^{2}+v_{0} t \text { for } t $$ (distance an object falls)

Exer. 53-56: Solve for the specified variable. $$ S=\pi r \sqrt{r^{2}+h^{2}} \text { for } h $$ (surface area of a cone)

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ |2 x+5|<4 $$

Exer. \(39-56:\) Find the solutions of the equation. $$ x^{3}+3 x^{2}+4 x=0 $$

When a hot gas exits a cylindrical smokestack, its velocity varies throughout a circular cross section of the smokestack, with the gas near the center of the cross section having a greater velocity than the gas near the perimeter. This phenomenon can be described by the formula $$ V=V_{\max }\left[1-\left(\frac{r}{r_{0}}\right)^{2}\right] $$ where \(V_{\max }\) is the maximum velocity of the gas, \(r_{0}\) is the radius of the smokestack, and \(V\) is the velocity of the gas at a distance \(r\) from the center of the circular cross section. Solve this formula for \(r\).

Determine values for \(a\) and \(b\) such that \(\frac{5}{3}\) is a solution of the equation. $$a x^{2}+b x=0$$

Exer. 53-56: Solve for the specified variable. \(\omega=\frac{1}{\sqrt{L C}}\) for \(C\) (alternating-current circuits)

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ |3 x-7| \geq 5 $$

Exer. \(39-56:\) Find the solutions of the equation. $$ 8 x^{3}-12 x^{2}+2 x-3=0 $$

For altitudes \(h\) up to 10,000 meters, the density \(D\) of Earth's atmosphere (in \(\mathrm{kg} / \mathrm{m}^{3}\) ) can be approximated by the formula $$ D=1.225-\left(1.12 \times 10^{-4}\right) h+\left(3.24 \times 10^{-9}\right) h^{2} . $$ Approximate the altitude if the density of the atmosphere is \(0.74 \mathrm{~kg} / \mathrm{m}^{3}\).

Determine which equation is not equivalent to the equation preceding it. $$ \begin{aligned} x^{2}-x-2 &=x^{2}-4 \\ (x+1)(x-2) &=(x+2)(x-2) \\ x+1 &=x+2 \\ 1 &=2 \end{aligned} $$

The recommended distance \(d\) that a ladder should be placed away from a vertical wall is \(25 \%\) of its length \(L\). Approximate the height \(h\) that can be reached by relating \(h\) as a percentage of \(L\). Exercise 57

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ -\frac{1}{3}|6-5 x|+2 \geq 1 $$

Exer. 57-62: Verify the property. $$ \overline{z+w}=\bar{z}+\bar{w} $$