Chapter 2

Exer. 71-72: Solve part (a) and use that answer to determine the answers to parts (b) and (c). (a) \(|x+5|=3\) (b) \(|x+5|<3\) (c) \(|x+5|>3\)

The formula occurs in the indicated application. Solve for the specified variable. \(s=\frac{1}{2} g t^{2}+v_{0} f\) for \(v_{0}\)

Exer. 71-72: Solve part (a) and use that answer to determine the answers to parts (b) and (c). (a) \(|x-3|<2\) (b) \(|x-3|=2\) (c) \(|x-3|>2\)

Two square wire frames are to be constructed from a piece of wire 100 inches long. If the area enclosed by one frame is to be one-half the area enclosed by the other, find the dimensions of each frame. (Disregard the thickness of the wire.)

The formula occurs in the indicated application. Solve for the specified variable. \(S=\frac{p}{q+p(1-q)}\) for \(q\)

Exer. 73-76: Express the statement in terms of an inequality involving an absolute value. The weight \(w\) of a wrestler must be within 2 pounds of 148 pounds.

The speed of the current in a stream is \(5 \mathrm{mi} / \mathrm{hr}\). It takes a canoeist 30 minutes longer to paddle \(1.2\) miles upstream than to paddle the same distance downstream. What is the canoeist's rate in still water?

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

Exer. 73-76: Express the statement in terms of an inequality involving an absolute value. The radius \(r\) of a ball bearing must be within \(0.01\) centimeter of 1 centimeter.

When a rock is dropped from a cliff into an ocean, it travels approximately \(16 t^{2}\) feet in \(t\) seconds. If the splash is heard 4 seconds later and the speed of sound is \(1100 \mathrm{ft} / \mathrm{sec}\), approximate the height of the cliff.

The formula occurs in the indicated application. Solve for the specified variable. \(\frac{1}{f}=\frac{1}{p}+\frac{1}{q}\) for \(q\)

Exer. 73-76: Express the statement in terms of an inequality involving an absolute value. The difference of two temperatures \(T_{1}\) and \(T_{2}\) within a chemical mixture must be between \(5^{\circ} \mathrm{C}\) and \(10^{\circ} \mathrm{C}\).

A company sells running shoes to dealers for $$\$ 40$$ per pair if less than 50 pairs are ordered. If 50 or more pairs are ordered (up to 600 ), the price per pair is reduced at a rate of $$\$ 0.04$$ times the number ordered. How many pairs can a dealer purchase for $$\$ 8400$$ ?

The formula occurs in the indicated application. Solve for the specified variable. \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}\) for \(R_{2}\)

Exer. 73-76: Express the statement in terms of an inequality involving an absolute value. The arrival time \(t\) of train B must be at least 5 minutes different from the 4:00 P.M. arrival time of train A.

When a popular brand of CD player is priced at $$\$ 300$$ per unit, a store sells 15 units per week. Each time the price is reduced by $$\$ 10$$, however, the sales increase by 2 per week. What selling price will result in weekly revenues of $$\$ 7000$$ ?

Temperature readings on the Fahrenheit and Celsius scales are related by the formula \(C=\frac{5}{9}(F-32)\). What values of \(F\) correspond to the values of \(C\) such that \(30 \leq C \leq 40\) ?

A closed right circular cylindrical oil drum of height 4 feet is to be constructed so that the total surface area is \(10 \pi \mathrm{ft}^{2}\). Find the diameter of the drum.

According to Hooke's law, the force \(F\) (in pounds) required to stretch a certain spring \(x\) inches beyond its natural length is given by \(F=(4.5) x\) (see the figure). If \(10 \leq F \leq 18\), what are the corresponding values for \(x\) ?

Ohm's law in electrical theory states that if \(R\) denotes the resistance of an object (in ohms), \(V\) the potential difference across the object (in volts), and \(I\) the current that flows through it (in amperes), then \(R=V / I\). If the voltage is 110 , what values of the resistance will result in a current that does not exceed 10 amperes?

Exer. 79-80: During a nuclear explosion, a fireball will be produced having a maximum volume \(V_{0}\). For temperatures below \(2000 \mathrm{~K}\) and a given explosive force, the volume \(V\) of the fireball \(t\) seconds after the explosion can be estimated using the given formula. (Note that the kelvin is abbreviated as \(\mathrm{K}\), not \({ }^{\circ} \mathrm{K}_{.}\)) Approximate \(t\) when \(V\) is \(95 \%\) of \(V_{0}\). \(\mathrm{~V} / V_{0}=0.8197+0.007752 t+0.0000281 t^{2}\) (20-kiloton explosion)

If two resistors \(R_{1}\) and \(R_{2}\) are connected in parallel in an electrical circuit, the net resistance \(R\) is given by $$ \frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}} \text {. } $$ If \(R_{1}=10\) ohms, what values of \(R_{2}\) will result in a net resistance of less than 5 ohms?

Exer. 81-82: When computations are carried out on a calculator, the quadratic formula will not always give accurate results if \(b^{2}\) is large in comparison to \(a c\), because one of the roots will be close to zero and difficult to approximate. (a) Use the quadratic formula to approximate the roots of the given equation. (b) To obtain a better approximation for the root near zero, rationalize the numerator to change $$ x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \text { to } x=\frac{2 c}{-b \mp \sqrt{b^{2}-4 a c}} $$ and use the second formula. $$ x^{2}+4,500,000 x-0.96=0 $$

To treat arrhythmia (irregular heartbeat), a drug is fed intravenously into the bloodstream. Suppose that the concentration \(c\) of the drug after \(t\) hours is given by \(c=3.5 t /(t+1) \mathrm{mg} / \mathrm{L}\). If the minimum therapeutic level is \(1.5 \mathrm{mg} / \mathrm{L}\), determine when this level is exceeded.

Exer. 81-82: When computations are carried out on a calculator, the quadratic formula will not always give accurate results if \(b^{2}\) is large in comparison to \(a c\), because one of the roots will be close to zero and difficult to approximate. (a) Use the quadratic formula to approximate the roots of the given equation. (b) To obtain a better approximation for the root near zero, rationalize the numerator to change $$ x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \text { to } x=\frac{2 c}{-b \mp \sqrt{b^{2}-4 a c}} $$ and use the second formula. $$ x^{2}-73,000,000 x+2.01=0 $$

A construction firm is trying to decide which of two models of a crane to purchase. Model A costs $$\$ 100,000$$ and requires $$\$ 8000$$ per year to maintain. Model B has an initial cost of $$\$ 80,000$$ and a maintenance cost of $$\$ 11,000$$ per year. For how many years must model A be used before it becomes more economical than B?

A consumer is trying to decide whether to purchase car A or car B. Car A costs $$\$ 20,000$$ and has an mpg rating of 30 , and insurance is $$\$ 1000$$ per year. Car B costs $$\$ 24,000$$ and has an mpg rating of 50 , and insurance is $$\$ 1200$$ per year. Assume that the consumer drives 15,000 miles per year and that the price of gas remains constant at \(\$ 3\) per gallon. Based only on these facts, determine how long it will take for the total cost of car B to become less than that of car A.

A person's height will typically decrease by \(0.024\) inch each year after age 30 . (a) If a woman was 5 feet 9 inches tall at age 30 , predict her height at age 70 . (b) A 50-year-old man is 5 feet 6 inches tall. Determine an inequality for the range of heights (in inches) that this man will experience between the ages of 30 and 70 .