91Ó°ÊÓ

Evaluate each expression using the graphs of y = f(x) and y = g(x) shown in the figure.

(a) (g o f)(-8)

(b) (fog)(-8)

(c) (g o g)(7)

(d) (g o f)(-5)

What is the interest due if $500 is borrowed for 6 months at a simple interest rate of 6% per annum?

A strain of E. coli Beu 397-recA441 is placed into a nutrient broth at 30° Celsius and allowed to grow. The following data are collected. Theory states that the number of bacteria in the petri dish will initially grow according to

the law of uninhibited growth. The population is measured using an optical device in which the amount of light that passes through the petri dish is measured.

(a) Draw a scatter diagram treating time as the independent variable.

(b) Using a graphing utility, build an exponential model from the data.

(c) Express the function found in part (b) in the form N(t) = N0 e^kt.

(d) Graph the exponential function found in part (b) or (c) on the scatter diagram.

(e) Use the exponential function from part (b) or (c) to predict the population at x = 7 hours.

(f) Use the exponential function from part (b) or (c) to predict when the population will reach 0.75.

Radioactive Decay

The half-life of radioactive potassium is 1.3 billion years. If 10 grams is present now, how much will be present in 100 years? In 1000 years?

$p\left(h\right)=760e^{-0.145h}$

(a) Find the atmospheric pressure at a height of 2 kilometers (over a mile).
(b) What is it at a height of 10 kilometers (over 30,000 feet)?

$F\left(t\right)=1-e^{-0.1t}$

(a) Determine the probability that a car will arrive within 10 minutes of 12:00 PM (that is, before 12:10 PM).
(b) Determine the probability that a car will arrive within 40 minutes of 12:00 PM (before 12:40 PM).
(c) What value doesFapproach astbecomes unbounded in the positive direction?
(d) Graph F using a graphing utility.
(e) Using INTERSECT, determine how many minutes are needed for the probability to reach $50\%$.

A piece of charcoal is found to contain 30% of the carbon 14 that it originally had.

(a) When did the tree from which the charcoal came die?

Use 5700 years as the half-life of carbon 14.

(b) Using a graphing utility, graph the relation between the percentage of carbon 14 remaining and time.

(c) Using INTERSECT, determine the time that elapses until half of the carbon 14 remains.

(d) Verify the answer found in part (a).

A fossilized leaf contains 70% of its normal amount of carbon 14.

(a) How old is the fossil? Use 5700 years as the half-life of carbon 14.

(b) Using a graphing utility, graph the relation between the percentage of carbon 14 remaining and time.

(c) Using INTERSECT, determine the time that elapses until half of the carbon 14 remains.

(d) Verify the answer found in part (a).

Current in a RL Circuit The equation governing the amount of current $I$(in amperes) after time $t$(in seconds) in a simple RL circuit consisting of a resistance $R$(in ohms), an inductance $L$(in henrys), and an electromotive force $E$(in volts) is

$I=\raisebox{1ex}{$E$}\!\left/ \!\raisebox{-1ex}{$R$}\right.\left[1-e^{-\left(\raisebox{1ex}{$R$}\!\left/ \!\raisebox{-1ex}{$L$}\right.\right)t}\right]$

If $E=12$volts, $R=10$ohms, and $L=5$henrys, how long does it take to obtain a current of $0.5$ ampere? Of $1.0$ ampere? Graph the equation.

The bacteria in a $4$-liter container double every minute. After $60$minutes the container is full. How long did it take to fill half the container?