Chapter 1

The admissions office of a college wants to determine whether there is a relationship between IQ scores \(x\) and grade-point averages \(y\) after the first year of school. An equation that models the data obtained by the admissions office is \(y=0.068 x-4.753\) Estimate the values of \(x\) that predict a grade-point average of at least \(3.0\).

MAKE A DECISION The temperature \(T\) (in degrees Fahrenheit) during a certain day can be approximated by \(T=0.31 t^{2}+32.9,7 \leq t \leq 15\), where \(t\) represents the hour of the day, with \(t=7\) corresponding to 7 A.M. Use the model to predict the time when the temperature was \(85^{\circ} \mathrm{F}\). Can you use this model to predict the temperature at 7 P.M.? Explain.

Solve for the indicated variable. Investment at Simple Interest Solve for \(r\) in \(A=P+P r t\).

You want to determine whether there is a relationship between an athlete's weight \(x\) (in pounds) and the athlete's maximum bench-press weight \(y\) (in pounds). An equation that models the data you obtained is \(y=1.4 x-39\) (a) Estimate the values of \(x\) that predict a maximum bench-press weight of at least 200 pounds. (b) Do you think an athlete's weight is a good indicator of the athlete's maximum bench-press weight? What other factors might influence an individual's bench-press weight?

Solve for the indicated variable. Investment at Compound Interest Solve for \(P\) in \(A=P\left(1+\frac{r}{n}\right)^{n t}\).

The average professional baseball player's salary \(S\) (in millions of dollars) from 1995 to 2006 can be modeled by \(S=0.1527 t+0.294, \quad 5 \leq t \leq 16\) where \(t\) represents the year, with \(t=5\) corresponding to 1995 (see figure). Use the model to predict the year in which the average professional baseball player's salary exceeds \(\$ 3,000,000\). (Source: Major League Baseball)

Blue Oak The blue oak tree, native to California, is known for its slow rate of growth. Fencing enclosures protect seedlings from herbivore damage and promote faster growth. The height \(H\) (in inches) of an enclosed blue oak tree can be approximated by the model \(H=0.74 t^{2}+25, \quad 0 \leq t \leq 5\) where \(t\) represents the year, with \(t=0\) corresponding to 2000\. Use the model to approximate the year in which the height of the tree was about 32 inches.

Solve for the indicated variable. Area of a Trapezoid Solve for \(b\) in \(A=\frac{1}{2}(a+b) h\).

Solve for the indicated variable. Area of a Sector of a Circle Solve for \(\theta\) in \(A=\frac{\pi r^{2} \theta}{360}\).

You measure the side of a square as \(10.4\) inches with a possible error of \(\frac{1}{16}\) inch. Using these measurements, determine the interval containing the possible areas of the square.

You measure the side of a square as \(24.2\) centimeters with a possible error of \(0.25\) centimeter. Using these measurements, determine the interval containing the possible areas of the square.

Solve for the indicated variable. Geometric Progression Solve for \(r\) in \(S=\frac{r L-a}{r-1}\).

You buy six T-bone steaks that cost \(\$ 7.99\) per pound. The weight listed on the package is \(5.72\) pounds. The scale that weighed the package is accurate to within \(\frac{1}{2}\) ounce. How much money might you have been undercharged or overcharged?

Solve for the indicated variable. Lateral Surface Area of a Cylinder Solve for \(h\) in \(A=2 \pi r h\).

You stop at a self-service gas station to buy 15 gallons of 87 -octane gasoline at \(\$ 2.42\) a gallon. The pump scale is accurate to within one-tenth of a gallon. How much money might you have been undercharged or overcharged?

Solve for the indicated variable. Surface Area of a Cone Solve for \(l\) in \(S=\pi r^{2}+\pi r l\).

The heights \(h\) of two-thirds of a population satisfy the inequality \(|h-68.5| \leq 2.7\) where \(h\) is measured in inches. Determine the interval on the real number line in which these heights lie.

Solve for the indicated variable. Lensmaker's Equation Solve for \(R_{1}\) in \(\frac{1}{f}=(n-1)\left(\frac{1}{R_{1}}-\frac{1}{R_{2}}\right)\)

A time study was conducted to determine the length of time required to perform a particular task in a manufacturing process. The times required by approximately two-thirds of the workers in the study satisfied the inequality \(\left|\frac{t-15.6}{1.9}\right|<1\) where \(t\) is time in minutes. Determine the interval on the real number line in which these times lie.

Solve for the indicated variable. Capacitance in Series Circuits Solve for \(C_{1}\) in \(C=\frac{1}{\frac{1}{C_{1}}+\frac{1}{C_{2}}}\)

The specifications for an electronic device state that it is to be operated in a room with relative humidity \(h\) defined by \(|h-50| \leq 30 .\) What are the minimum and maximum relative humidities for the operation of this device?

Physicians consider an adult's body temperature \(x\) (in degrees Fahrenheit) to be normal if it satisfies the inequality \(|x-98.6| \leq 1\) Determine the range of temperatures that are considered to be normal.

The average price \(B\) (in dollars) of brand name prescription drugs from 1998 to 2005 can be modeled by \(B=6.928 t-3.45, \quad 8 \leq t \leq 15\) where \(t\) represents the year, with \(t=8\) corresponding to 1998 . Use the model to find the year in which the price of the average brand name drug prescription exceeded \(\$ 75\).

The average price \(G\) (in dollars) of generic prescription drugs from 1998 to 2005 can be modeled by \(G=2.005 t+0.40, \quad 8 \leq t \leq 15\) where \(t\) represents the year, with \(t=8\) corresponding to \(1998 .\) Use the model to find the year in which the price of the average generic drug prescription exceeded \(\$ 19\).

The daily demand \(D\) (in thousands of barrels) for refined oil in the United States from 1995 to 2005 can be modeled by \(D=276.4 t+16,656, \quad 5 \leq t \leq 15\) where \(t\) represents the year, with \(t=5\) corresponding to 1995. (a) Use the model to find the year in which the demand for U.S. oil exceeded 18 million barrels a day. (b) Use the model to predict the year in which the demand for U.S. oil will exceed 22 million barrels a day.

The daily amount \(I\) (in thousands of barrels) of crude oil imported to the United States from 1995 to 2005 can be modeled by \(I=428.2 t+6976, \quad 5 \leq t \leq 15\) where \(t\) represents the year, with \(t=5\) corresponding to 1995. (a) Use the model to find the year in which the amount of crude oil imported to the United States exceeded 12 million barrels a day. (b) Use the model to predict the year in which the amount of oil imported to the United States will exceed 14 million barrels a day.