Exam 5: Applications of the Exponential and Natural Logarithm Functions

Find a so that y = 1 - a $\mathrm { e } ^ { - \mathrm { x } }$ solves the differential equation $y ^ { \prime } = y - 1 + 6 e ^ { - x }$ Enter just an integer.

A savings account pays 7% interest, compounded continuously. How much should be deposited now in order to have $5000 in the account at the end of five years?

Plutonium 239 has a half-life of 24,000 years. What is its decay constant? Enter your answer as just a real number to eight decimal places.

What rate of interest is required in order for a $100 investment to double in 3 years if the interest is compounded continuously?

It is observed that the sales of a certain recording fall to 75% of their original level one month after advertising stops. If this continues, what will be the sales after 4 months? Enter just a real number rounded to one decimal place (no units or symbols).

Mr. Jones has two investments. The first is currently worth $50,000 and has an annual yield of 10% compounded continuously. The second is currently worth $70,000 and has an annual yield of 8% compounded continuously. After how many years will the two investments be worth the same amount? Enter your answer in years to one decimal place (no units).

Let P(t) be the quantity of strontium-90 remaining after t years. Suppose the half-life of strontium-90 is 28 years. Which of the following equations expresses the half-life information?

Krypton 85 gas leaks into the reactor room of an electric power plant. Its half-life is 10 years. How long is it before 99.9% of the krypton decays? Enter your answer as just a real number rounded to one decimal place (no units).

Suppose that a school of fish in a pond grows according to the exponential law $P ( t ) = P 0 e ^ { k t }$ and suppose that the size of the colony triples in 24 days. If the initial size of the school was 50, when will the school contain 200 fish? Enter your answer exactly in the form $\frac { \mathrm { a } \ln \mathrm { b } } { \ln \mathrm { c } }$

The size of an insect colony t days after its formation is P( t) = 1000 $e ^ { 0.2 t }$ . Approximately how many insects are present after 10 days?

Suppose a manufacturer can sell q = $\frac { 1000 } { ( p + 2 ) ^ { 2 } }$ - 6 units of a product when the price is p dollars per unit. Determine the elasticity of demand, E(p), when the price is p = 8 dollars. Enter just an integer.

A country has a population of 287 million in 2005. Assuming a growth rate of 1.3%, determine the function that expresses the population of the country t years after 2005.

A high-yield savings pays 20% interest, compounded continuously. How long would it take an initial investment of $2500 to grow to $20,000?

The population of a colony of bacteria triples in 3 days. Assuming that the rate of growth is proportional to the size of the population, how long did it take for the colony to double in size?

A radioactive substance is observed to disintegrate at a rate such that $\frac { 9 } { 10 }$ of the original amount remains after one year. What is the half-life of the substance?

What rate of interest will make an investment triple in 8 years if the interest is compounded continuously? Enter your answer as a percent to one decimal place $\text { (no } \% \text { symbol or words) }$

Carbon 14 has a half-life of 5730 years. Determine its decay constant. Enter just a real number rounded to five decimal places.

A certain radioactive element has a half-life of 12 minutes. At what time is the substance decaying at a rate of 3.466 grams per minute if there are 120 grams present initially?

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 approximately what rate will the rumor be spreading after 7 days? Enter just an integer rounded to the nearest hundred (no words or units).

Determine the percentage rate of change of h(p) = $\frac { 4 } { 7 p + 3 }$ at p = 1 and p = 9.