Exam 5: Applications of the Exponential and Natural Logarithm Functions

A colony of bacteria is growing at a rate proportional to the number of bacteria present. At the beginning of an experiment there were about $10 ^ { 3 }$ bacteria present. In two hours, the count rose to $3 \times 10 ^ { 3 }$ bacteria. At what time will there be 6 × $10 ^ { 3 }$ bacteria present? Enter just a real number rounded up to one decimal place (no units).

A parchment is offered for sale at a Paris flea market. The owner claims it is at least 2000 years old. However, a carbon-dating test shows that ${ } ^ { 14 } \mathrm { C }$ - ${}^{12} \mathrm { C }$ ratio for the manuscript is 95% of the corresponding ratio for currently manufactured parchment. How old is the manuscript? (The decay constant of ${ } ^ { 14 } \mathrm { C }$ is 0.00012)Enter your answer exactly in the form $\frac { \ln a } { \mathrm {~b} }$ (no units).

Suppose that $500 is deposited in a bank certificate paying 10% interest compounded continuously. How much will the certificate be worth after 5 years? Enter your answer in dollars to two decimal places (no symbols or words).

A savings account paying 3% continuously compounded interest has $1600 after 8 years. How much was initially deposited?

A high school student deposits a $500 graduation gift in a bank account that pays 4.8% interest compounded continuously. How much will the account be worth after 18 months?

How long is required for an investment of $2000 to double if the interest is 10% compounded continuously? Enter your answer in years to one decimal place $\text { (no units) }$

Let A'(t) = 0.3(25,000 - A(t)), A(0) = 0. Which of the following is the formula for A(t)?

If the relative rate of change of a function f is always 5, what kind of equation will f be represented by? Enter your answer in the form: f(t) = C $\mathrm { e } ^ { \mathrm { b } }$ (leave C in your answer).

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. -At what rate will the rumor be spreading when half of the potential audience has heard the rumor? Enter just an integer rounded to the nearest hundred (no units).

A company has a demand function q= 200 - 40p. Determine the elasticity of demand as a function of p. Enter your answer exactly as $\frac{P ( p )}{Q ( p )}$ where P, Q are polynomials in p in standard form.

Suppose that at any time t, a colony of fruit flies is growing at a rate equal to one half the current size of the colony. Find a formula which gives the size of the colony at time t if there were originally 500 fruit flies present. Enter your answer exactly in the form: $\mathrm { P } ( \mathrm { t } ) = a \mathrm { e } ^ { \mathrm { b } t }$

Determine the percentage rate of change of f(x) = $e ^ { 0.9 x }$ at x = 15 and x = 30.

Eight years ago, $2000 was deposited in a savings account paying 3% interest compounded continuously. Three years ago, $500 was withdrawn from the account. What is the current value of the account?