Chapter 2

A manufacturer of tin cans wishes to construct a right circular cylindrical can of height 20 centimeters and capacity \(3000 \mathrm{~cm}^{3}\) (see the figure). Find the inner radius \(r\) of the can.

Determine which equation is not equivalent to the equation preceding it. $$ \begin{aligned} 5 x+6 &=4 x+3 \\ x^{2}+5 x+6 &=x^{2}+4 x+3 \\ (x+2)(x+3) &=(x+1)(x+3) \\ x+2 &=x+1 \\ 2 &=1 \end{aligned} $$

Nuclear experiments performed in the ocean vaporize large quantities of salt water. Salt boils and turns into vapor at \(1738 \mathrm{~K}\). After being vaporized by a 10 -megaton force, the salt takes at least \(8-10\) seconds to cool enough to crystallize. The amount of salt \(A\) that has crystallized \(t\) seconds after an experiment is sometimes calculated using \(A=k \sqrt{t / T}\), where \(k\) and \(T\) are constants. Solve this equation for \(t\).

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ 2|-11-7 x|-2>10 $$

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

Solve the formula for the specified variable. \(E K+L=D-T K\) for \(K\)

The power \(P\) (in watts) generated by a windmill that has efficiency \(E\) is given by the formula \(P=0.31 E D^{2} V^{3}\), where \(D\) is the diameter (in feet) of the windmill blades and \(V\) is the wind velocity (in \(\mathrm{ft} / \mathrm{sec}\) ). Approximate the wind velocity necessary to generate 10,000 watts if \(E=42 \%\) and \(D=10\).

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ |7 x+2|>-2 $$

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

A baseball is thrown straight upward with an initial speed of \(64 \mathrm{ft} / \mathrm{sec}\). The number of feet \(s\) above the ground after \(t\) seconds is given by the equation \(s=-16 t^{2}+64 t\) (a) When will the baseball be 48 feet above the ground? (b) When will it hit the ground?

Solve the formula for the specified variable. \(C D+C=P C+N\) for \(C\)

The withdrawal resistance of a nail indicates its holding strength in wood. A formula that is used for bright common nails is \(P=15,700 S^{5 / 2} R D\), where \(P\) is the maximum withdrawal resistance (in pounds), \(S\) is the specific gravity of the wood at \(12 \%\) moisture content, \(R\) is the radius of the nail (in inches), and \(D\) is the depth (in inches) that the nail has penetrated the wood. A \(6 \mathrm{~d}\) (sixpenny) bright, common nail of length 2 inches and diameter \(0.113\) inch is driven completely into a piece of Douglas fir. If it requires a maximum force of 380 pounds to remove the nail, approximate the specific gravity of Douglas fir.

The distance that a car travels between the time the driver makes the decision to hit the brakes and the time the car actually stops is called the braking distance. For a certain car traveling \(v \mathrm{mi} / \mathrm{hr}\), the braking distance \(d\) (in feet) is given by \(d=v+\left(v^{2} / 20\right)\). (a) Find the braking distance when \(v\) is \(55 \mathrm{mi} / \mathrm{hr}\). (b) If a driver decides to brake 120 feet from a stop sign, how fast can the car be going and still stop by the time it reaches the sign?

Solve the formula for the specified variable. $$M=\frac{Q+1}{Q}\( for \)Q$$

The demand for a commodity usually depends on its price. If other factors do not affect the demand, then the quantity \(Q\) purchased at price \(P\) (in cents) is given by \(Q=k P^{-c}\), where \(k\) and \(c\) are positive constants. If \(k=10^{5}\) and \(c=\frac{1}{2}\), find the price that will result in the purchase of 5000 items.

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ |3 x-9|>0 $$

Exer. 57-62: Verify the property. $$ \bar{z}=z \text { if and only if } z \text { is real. } $$

The temperature \(T\) (in \({ }^{\circ} \mathrm{C}\) ) at which water boils is related to the elevation \(h\) (in meters above sea level) by the formula $$ h=1000(100-T)+580(100-T)^{2} $$ for \(95 \leq T \leq 100\). (a) At what elevation does water boil at a temperature of \(98^{\circ} \mathrm{C}\) ? (b) The elevation of Mt. Everest is approximately 8840 meters. Estimate the temperature at which water boils at the top of this mountain. (Hint: Use the quadratic formula with \(x=100-T\).)

Solve the formula for the specified variable. $$\beta=\frac{\alpha}{1-\alpha}\( for \)\alpha$$

Urban areas have higher average air temperatures than rural areas, as a result of the presence of buildings, asphalt, and concrete. This phenomenon has become known as the urban heat island. The temperature difference \(T\) (in \({ }^{\circ} \mathrm{C}\) ) between urban and rural areas near Montreal, with a population \(P\) between 1000 and \(1,000,000\), can be described by the formula \(T=0.25 P^{1 / 4} / \sqrt{v}\), where \(v\) is the average wind speed (in mi/hr) and \(v \geq 1\). If \(T=3\) and \(v=5\), find \(P\).

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ |5 x+2| \leq 0 $$

A particle of charge \(-1\) is located on a coordinate line at \(x=-2\), and a particle of charge \(-2\) is located at \(x=2\), as shown in the figure. If a particle of charge \(+1\) is located at a position \(x\) between \(-2\) and 2 , Coulomb's law in electrical theory asserts that the net force \(F\) acting on this particle is given by $$ F=\frac{-k}{(x+2)^{2}}+\frac{2 k}{(2-x)^{2}} $$ for some constant \(k>0\). Determine the position at which the net force is zero.

The formula occurs in the indicated application. Solve for the specified variable. \(I=P r t\) for \(P\)

As sand leaks out of a certain container, it forms a pile that has the shape of a right circular cone whose altitude is always one-half the diameter \(d\) of the base. What is \(d\) at the instant at which \(144 \mathrm{~cm}^{3}\) of sand has leaked out? Exercise 63

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

A rectangular plot of ground having dimensions 26 feet by 30 feet is surrounded by a walk of uniform width. If the area of the walk is \(240 \mathrm{ft}^{2}\), what is its width?

The formula occurs in the indicated application. Solve for the specified variable. \(C=2 \pi r\) for \(r\)

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

A 24-by-36-inch sheet of paper is to be used for a poster, with the shorter side at the bottom. The margins at the sides and top are to have the same width, and the bottom margin is to be twice as wide as the other margins. Find the width of the margins if the printed area is to be \(661.5 \mathrm{in}^{2}\).

The formula occurs in the indicated application. Solve for the specified variable. \(A=\frac{1}{2} b h\) for \(h\)

The cube rule in political science is an empirical formula that is said to predict the percentage \(y\) of seats in the U.S. House of Representatives that will be won by a political party from the popular vote for the party's presidential candidate. If \(x\) denotes the percentage of the popular vote for a party's presidential candidate, then the cube rule states that $$ y=\frac{x^{3}}{x^{3}+(1-x)^{3}} $$ What percentage of the popular vote will the presidential candidate need in order for the candidate's party to win \(60 \%\) of the House seats?

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

A square vegetable garden is to be tilled and then enclosed with a fence. If the fence costs \(\$ 1\) per foot and the cost of preparing the soil is \(\$ 0.50\) per \(\mathrm{ft}^{2}\), determine the size of the garden that can be enclosed for \(\$ 120\).

The formula occurs in the indicated application. Solve for the specified variable. \(V=\frac{1}{3} \pi r^{2} h\) for \(h\)

A conical paper cup is to have a height of 3 inches. Find the radius of the cone that will result in a surface area of \(6 \pi\) in \(^{2}\).

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

A farmer plans to enclose a rectangular region, using part of his barn for one side and fencing for the other three sides. If the side parallel to the barn is to be twice the length of an adjacent side, and the area of the region is to be \(128 \mathrm{ft}^{2}\), how many feet of fencing should be purchased?

The formula occurs in the indicated application. Solve for the specified variable. \(F=g \frac{m M}{d^{2}}\) for \(m\)

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

The formula occurs in the indicated application. Solve for the specified variable. \(R=\frac{V}{I}\) for \(I\)

Calculating human growth Adolphe Quetelet (1796-1874), the director of the Brussels Observatory from 1832 to 1874 , was the first person to attempt to fit a mathematical expression to human growth data. If \(h\) denotes height in meters and \(t\) denotes age in years, Quetelet's formula for males in Brussels can be expressed as $$ h+\frac{h}{h_{M}-h}=a t+\frac{h_{0}+t}{1+\frac{4}{3} t} $$ with \(h_{0}=0.5\), the height at birth; \(h_{M}=1.684\), the final adult male height; and \(a=0.545\). (a) Find the expected height of a 12-year-old male. (b) At what age should \(50 \%\) of the adult height be reached?

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ 1<|x|<5 $$

The boundary of a city is a circle of diameter 10 miles. Within the last decade, the city has grown in area by approximately \(16 \pi \mathrm{mi}^{2}\) (about \(50 \mathrm{mi}^{2}\) ). Assuming the city was always circular in shape, find the corresponding change in distance from the center of the city to the boundary.

The formula occurs in the indicated application. Solve for the specified variable. \(P=2 l+2 w\) for \(w\)

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ 1<|x-2|<4 $$

An airplane flying north at \(200 \mathrm{mi} / \mathrm{hr}\) passed over a point on the ground at 2:00 P.M. Another airplane at the same altitude passed over the point at 2:30 P.M., flying east at \(400 \mathrm{mi} / \mathrm{hr}\) (see the figure). (a) If \(t\) denotes the time in hours after 2:30 P.M., express the distance \(d\) between the airplanes in terms of \(t\). (b) At what time after 2:30 P.M. were the airplanes 500 miles apart?

The formula occurs in the indicated application. Solve for the specified variable. \(A=P+P r\) for \(r\)

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ 2<|2 x-1|<3 $$

Two surveyors with two-way radios leave the same point at 9:00 A.M., one walking due south at \(4 \mathrm{mi} / \mathrm{hr}\) and the other due west at \(3 \mathrm{mi} / \mathrm{hr}\). How long can they communicate with one another if each radio has a maximum range of 2 miles?

The formula occurs in the indicated application. Solve for the specified variable. \(A=\frac{1}{2}\left(b_{1}+b_{2}\right) h\) for \(b_{1}\)