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Carbon \( 14 \) dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true,the amount of \( ^{14} C \) absorbed by a tree that grew several centuries ago should be the same as the amount of \( ^{14} C \) absorbed by a tree growing today. A piece of ancient charcoal contains only \( 15\% \) as much radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal if the half-life of \( ^{14} C \) is \( 5715 \) years?

A sport utility vehicle that costs \( \$23,300 \) new has a book value of \( \$12,500 \) after \( 2 \) years. (a) Find the linear model \( V = mt + b \). (b) Find the exponential model \( V = ae^{kt} \). (c) Use a graphing utility to graph the two models in the same viewing window. Which model depreciates faster in the first \( 2 \) years? (d) Find the book values of the vehicle after \( 1 \) year and after \( 3 \) years using each model. (e) Explain the advantages and disadvantages of using each model to a buyer and a seller.

A cell site is a site where electronic communications equipment is placed in a cellular network for the use of mobile phones. The numbers of cell sites from \( 1985 \) through \( 2008 \) can be modeled by \( y = \dfrac{237,101}{1 + 1950e^{-0.355t} \) where \( t \) represents the year, with \( t = 5 \) corresponding to \( 1985 \).(Source: CTIA-The Wireless Association) (a) Use the model to find the numbers of cell sites in the years \( 1985 \), \( 2000 \), and \( 2006 \). (b) Use a graphing utility to graph the function. (c) Use the graph to determine the year in which the number of cell sites will reach \( 235,000 \). (d) Confirm your answer to part (c) algebraically.

Due to the installation of noise suppression materials,the noise level in an auditorium was reduced from \( 93 \) to \( 80 \) decibels. Find the percent decrease in the intensity level of the noise as a result of the installation of these materials.

If the annual rate of inflation averages \( 4\% \) over the next \( 10 \) years, the approximate costs \( C \) of goods or services during any year in that decade will be modeled by \( C(t) = P(1.04)^t \), where \( t \) is the time in years and \( P \) is the present cost. The price of an oil change for your car is presently \( \$23.95 \). Estimate the price \( 10 \) years from now.

In Exercises 69 - 74, use the acidity model given by \( pH = -\log \left[H^+\right] \), where acidity \( (pH) \) is a measure of the hydrogen ion concentration \( \left[H^+\right] \) (measured in moles of hydrogen per liter) of a solution. Compute \( \left[H^+\right] for a solution in which \) pH = 3.2. $

Apple juice has a \( pH \) of \( 2.9 \) and drinking water has a \( pH \) of \( 8.0. \) The hydrogen ion concentration of the apple juice is how many times the concentration of drinking water?

The \( pH \) of a solution is decreased by one unit. The hydrogen ion concentration is increased by what factor?

At \( 8:30 \) A.M., a coroner was called to the home of a person who had died during the night. In order to estimate the time of death, the coroner took the persons temperature twice. At \( 9:00 \) A.M. the temperature was \( 85.7^\circ F \) and at \( 11:00 \) A.M. the temperature was \( 82.8^\circ F \). From these two temperatures,the coroner was able to determine that the time elapsed since death and the body temperature were related by the formula \( t = -10 ln \dfrac{T - 70}{98.6 - 70} where \) t \( is the time in hours elapsed since the person died and \) T \( is the temperature (in degrees Fahrenheit) of the persons body. (This formula is derived from a general cooling principle called Newtons Law of Cooling. It uses the assumptions that the person had a normal body temperature of \) 98.6^\circ F \( at death, and that the room temperature was a constant \) 70^\circ F $. ) Use the formula to estimate the time of death of the person.

In Exercises 79 - 82, determine whether the statement is true or false. Justify your answer. The graph of a Gaussian model will never have an \( x \)-intercept.