Exam 5: Applications of the Exponential and Natural Logarithm Functions

For the demand function q = $p ^ { 2 }$ $e ^ { - ( p + 7 ) }$ , find E(p) and determine if the demand is elastic or inelastic (or neither) at the price p = 5.

Radioactive carbon 11 has a half-life of 20 minutes. If there are 200 grams present at the start of our experiment, how many grams will remain after 10 minutes?

In a town of 10,000 people, the number of people who during each day first hear the news of a local tax increase is one-tenth the number of people who have not yet heard the news. If f(t) stands for the number of informed people in the town, what is the differential equation f(t) satisfies?

Barium 140 has a half-life of 13 days. After 25 days a given sample comprises 5 grams. How large was the original sample? Enter your answer as a real number to two decimal places (no units).

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 1250 people have heard the rumor? Enter just an integer (no units).

The decay constant for strontium 90 is λ = 0.0244, where the time is measured in years. How long will it take for a quantity $\mathrm { P } _ { 0 }$ of strontium 90 to decay to $\frac { 1 } { 3 }$ its original size? Enter your answer exactly in the form $\frac { \ln a } { \mathrm {~b} }$ where a is a reduced fraction and b is a real number to four decimal places (no units).

$1,000 is invested at r% interest continuously compounded. After 6 years the account has reached $1350. What was the interest rate?

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 how many people will have heard the rumor after 7 days? Enter your answer as just an integer rounded to the nearest hundred.

You deposit $500 in a savings account paying 4% interest continuously compounded. How much will the account be worth after 32 months?

Potassium has a half-life of 12 hours. How long will it take for a quantity of potassium to decay to $\frac { 1 } { 10 }$ its original size? Enter your answer as a real number rounded to one decimal place (no units).

Suppose that the value in billions of dollars of a company is determined to be $f ( t ) = 0.5 t + 0.2 e ^ { - t }$ where t is measured in years. What is the percentage rate of growth of the company at time t = 0? Enter just an integer (no units).

What is the present value of an investment of $1000 payable at the end of 10 years at a 9% rate of interest compounded continuously? Enter your answer in dollars to two decimal places (no symbols or words).

A manufacturer has a demand function q = $\sqrt { 100 - p ^ { 3 } }$ . 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, and the leading coefficient of P is a reduced fraction of form $\frac { a } { b }$ .

How long will it take for an investment to triple if interest is paid at 10%, compounded continuously?

The population of a certain region was 10 million in 1950. By 1970, it had increased to 13.5 million. Assuming exponential growth, estimate the population in the year 2000. Enter your answer exactly in the form a $\mathrm { e } ^ { \mathrm { C } }$ (no units).

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. -When will the rumor be spreading at the greatest rate? Enter just an integer (round to the nearest integer, no units)

Find a so that y = 1 + $e ^ { a x ^ { 2 } }$ solves the differential equation $y ^ { \prime } = x - x y$ Enter just a reduced fraction of form $\frac { a } { b }$ .

In a certain country, the rate of increase of the population is proportional to the population P(t). In fact, $P ^ { \prime } ( t ) = 0.23 P ( t )$ Suppose that initially the country's population is 50,000, and that 10 years later there are 500,000 people. Which of the following equations expresses this information mathematically?

A bacterial culture grows exponentially; that is, P(t) = 100 $\mathrm { e^{kt} }$ , where P(t) is the size of the culture at time t hours. Suppose that after 2 hours the size of the culture is 400. What is k (approximately)?

Determine the percentage rate of change of f(t) = 2 $t ^ { 3 }$ at t = 5. Enter your answer as just a reduced fraction of form $\frac { a } { b }$ .