Exam 5: Applications of the Exponential and Natural Logarithm Functions

A fossil was discovered that had about 70% of the ${ } ^ { 14 } \mathrm { C }$ level found today in living matter. Given that the decay constant for ${ } ^ { 14 } \mathrm { C }$ is 0.00012, determine the age of the fossil. Enter your answer exactly as just $\frac { \ln a } { \mathrm {~b} }$ where a is a real number to one decimal place and b is a real number to 5 decimal places.

Assume that a culture of bacteria grows at a rate proportional to its size such that if $10 ^ { 6 }$ bacteria are present initially, then there are 2 × $10 ^ { 6 }$ bacteria present after 3 hours. Determine a formula for the number of bacteria present after t hours in terms of powers of 2. (Hint: Recall that $b ^ { x }$ = $\mathrm { e } ^ { ( \ln b ) x }$ .)Enter your answer exactly in the form $\mathrm { P } ( \mathrm { x } ) = \mathrm { a }^ b \cdot 2 ^{\mathrm { c } / \mathrm { d }}$

Sixteen pounds of a radioactive substance loses one fourth of its original mass in 2 days. The mass m(t) remaining at time t is given by:

Plutonium has a decay rate of 0.003% per year. What is the half life?

For the demand function q = 150(245 - $p ^ { 2 }$ ), find E(p) and determine if the demand is elastic or inelastic (or neither) at the price p =7.

A rumor is spread by word of mouth to a potential audience of 3000 people. After t days, $f ( t ) = \frac { 3000 } { 1 + 25 e ^ { - 0.7 t } }$ people will have heard the rumor. The graphs are shown below. -Approximately when will the rumor be spreading at a rate of 200 people per day? Enter just two integers a, b where a< (round to the nearest integers).

Suppose that the value of a certain investment after t years can be approximated by the function $f ( t ) = 100,000 e ^ { 0.12 t ^ { 2 / 3 } }$ What is the dollar value of the investment after 8 years? Enter your answer exactly in the form: a $\mathrm { e } ^ { \mathrm { b } }$ where a is an integer and b is a real number to two decimal places (no units).

Which of the following functions satisfy the differential equation y' = 3(12 - y)? (I) y = 4(3 - $e ^ { - 3 x }$ ) (II) y = 12(1 - $e ^ { - 3 x }$ ) (III) y = 12 - $e ^ { - 3 x }$ (IV) y = 3(1 - $e ^ { - 12 x }$ )

${ } ^ { 14 } \mathrm { C }$ has a half life of 5730 years. How old is a piece of charcoal which has lost 90% of its ${ } ^ { 14 } \mathrm { C }$ ?

Determine the percentage rate of change of F(t) = $\mathrm { e } ^ { - 0.09 \mathrm { t } ^ { 2 } }$ at t = 1 and t = 5.

Let g(t) = 100 - 100 $e ^ { - 0.01 t }$ be the number of cases of measles in a certain school t days after the first case is reported. Which of the following best describes the spread of the disease?

If y = 10(1 - $e ^ { - 2 x }$ ) and y' = a -2y, find a. Enter just an integer.

If y = 2(1 - $e ^ { - 4 x }$ ) and y' = 4 - by, find b. Enter just an integer.

Suppose that a school of fish in a pond grows according to the exponential law $\mathrm { P } ( \mathrm { t } ) = \mathrm { P } _ { 0 } \mathrm { e } ^ { \mathrm { kt } }$ and suppose that the size of the colony triples in 24 days. Determine k. Enter your answer exactly in the form $\frac { \ln a } { \mathrm {~b} }$ where a, b are integers.

A certain radioactive substance is decaying at a rate proportional to the amount present. If 100 grams decays to 13.5 grams in 4 years, how long will it take for 90 grams to decay to 30 grams?

$1000 is invested at 6% interest compounded continuously. What is the value of the investment after 5 years?

A bank pays 2.5% interest on deposits. What is the return on a $1000 deposit after two years if interest is compounded continuously?

How much money has to be invested now at 8% continuous interest in order to have $1000 after 5 years?

If an investment doubles in 5 years continuously compounded, how long will it take for the investment to triple?

Let y = 6(1 - $e ^ { - 3 x }$ ). What is 3(6 - y) ?