Question 1

Choose the one alternative that best completes the statement or answers the question.
-The logistic growth function $\mathrm { f } ( \mathrm { t } ) = \frac { 440 } { 1 + 7.8 \mathrm { e } ^ { - 0.17 \mathrm { t } } }$ describes the population of a species of butterflies $\mathrm { t }$ months after they are introduced to a non-threatening habitat. How many butterflies were initially introduced to the habitat?
A) 2 butterflies
B) 50 butterflies
C) 8 butterflies
D) 440 butterflies

Accepted Answer

The initial population is found by evaluating the logistic growth function at $t = 0$. Substituting $t = 0$ into the function gives $\mathrm { f } ( 0 ) = \frac { 440 } { 1 + 7.8 \mathrm { e } ^ { - 0.17 \times 0 } } = \frac { 440 } { 1 + 7.8 } = \frac { 440 } { 8.8 } = 50$. Therefore, 50 butterflies were initially introduced to the habitat.

Question 2

Write the word or phrase that best completes each statement or answers the question.
-A thermometer is taken from a room at $71 ^ { \circ } \mathrm { F }$ to the outdoors where the temperature is $14 ^ { \circ } \mathrm { F }$. Determine what the reading on the thermometer will be after 5 minutes, if the reading drops to $45 ^ { \circ } \mathrm { F }$ after 1 minute. Assume the cooling follows Newton's Law of Cooling:
$$\mathrm { U } = \mathrm { T } + \left( \mathrm { U } _ { \mathrm { O } } - \mathrm { T } \right) \mathrm { e } ^ { \mathrm { kt } } .$$
(Round your answer to two decimal places.)

Accepted Answer

The answer of Write the word or phrase that best...

Question 3

Choose the one alternative that best completes the statement or answers the question.
-The logistic growth model $\mathrm { P } ( \mathrm { t } ) = \frac { 900 } { 1 + 21.5 \mathrm { e } ^ { - 0.352 \mathrm { t } } }$ represents the population of a bacterium in a culture tube after $t$ hours. When will the amount of bacteria be 760 ?
A) $7.84$ hours
B) $2.55$ hours
C) $8.24$ hours
D) $13.52$ hours

Accepted Answer

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Question 4

Choose the one alternative that best completes the statement or answers the question.
-In 1990, the population of a country was estimated at 4 million. For any subsequent year the population, $\mathrm { P } ( \mathrm { t } )$ (in millions), can be modeled by the equation $\mathrm { P } ( \mathrm { t } ) = \frac { 240 } { 5 + 54.99 \mathrm { e } ^ { - 0.0208 \mathrm { t } } }$, where $t$ is the number of years since 1990 .
Estimate the year when the population will be 21 million.
A) approximately the year 2088
B) approximately the year 2016
C) approximately the year 2093
D) approximately the year 2041

Accepted Answer

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Question 5

Choose the one alternative that best completes the statement or answers the question.
-A thermometer reading $11 ^ { \circ } \mathrm { C }$ is brought into a room with a constant temperature of $30 ^ { \circ } \mathrm { C }$. If the thermometer reads $17 ^ { \circ } \mathrm { C }$ after 4 minutes, what will it read after being in the room for 7 minutes? Assume the cooling follows Newton's Law of Cooling:
$$\mathrm { U } = \mathrm { T } + \left( \mathrm { U } _ { \mathrm { O } } - \mathrm { T } \right) \mathrm { e } ^ { \mathrm { kt } } \text {. }$$
(Round your answer to two decimal places.)
A) $28.67 ^ { \circ } \mathrm { C }$
B) $5.87 ^ { \circ } \mathrm { C }$
C) $20.22 ^ { \circ } \mathrm { C }$
D) $39.78 ^ { \circ } \mathrm { C }$

Accepted Answer

Using Newton's Law of Cooling, solve for the constant $k$ with the given temperatures at 4 minutes, then use it to find the temperature at 7 minutes.

Question 6

Choose the one alternative that best completes the statement or answers the question.
-The logistic growth model $\mathrm { P } ( \mathrm { t } ) = \frac { 1 } { 1 + 6.69 \mathrm { e } ^ { - 0.753 \mathrm { t } } }$ represents the proportion of the total market of a new product as it penetrates the market t years after introduction. When will the product have $65 \%$ of the market?
A) $2.95$ years
B) $3.35$ years
C) $4.35$ years
D) $1.95$ years

Accepted Answer

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Question 7

Choose the one alternative that best completes the statement or answers the question.
-In a town whose population is 3000 , a disease creates an epidemic. The number of people, $N$, infected $t$ days afte: disease has begun is given by the function $\mathrm { N } ( \mathrm { t } ) = \frac { 3000 } { 1 + 21.2 \mathrm { e } ^ { - 0.54 t } }$. Find the number of infected people after 10 days.
A) 142 people
B) 2000 people.
C) 1000 people
D) 2737 people

Accepted Answer

To find the number of infected people after 10 days, substitute $t = 10$ into the function: $N(10) = \frac{3000}{1 + 21.2e^{-0.54 \times 10}}$. Calculating this gives approximately 2737 people.

Question 8

Write the word or phrase that best completes each statement or answers the question.
-A fully cooked turkey is taken out of an oven set at $200 ^ { \circ } \mathrm { C }$ (Celsius) and placed in a sink of chilled water of temperature $4 ^ { \circ } \mathrm { C }$. After 3 minutes, the temperature of the turkey is measured to be $50 ^ { \circ } \mathrm { C }$. How long (to the nearest minute) will it take for the temperature of the turkey to reach $15 ^ { \circ } \mathrm { C }$ ? Assume the cooling follows Newton Law of Cooling:
$$\mathrm { U } = \mathrm { T } + \left( \mathrm { U } _ { \mathrm { O } } - \mathrm { T } \right) \mathrm { e } ^ { \mathrm { kt } } .$$
(Round your answer to the nearest minute.)

Accepted Answer

The answer of Write the word or phrase that best...

Question 9

Choose the one alternative that best completes the statement or answers the question.
-The logistic growth model $\mathrm { P } ( \mathrm { t } ) = \frac { 410 } { 1 + 40 \mathrm { e } ^ { - 0.176 \mathrm { t } } }$ represents the population of a species introduced into a new territory after $t$ years. When will the population be 60 ?
A) $10.94$ years
B) $6.34$ years
C) $3.69$ years
D) $16.76$ years

Accepted Answer

The population will be 60 when $P(t) = 60$. Solving this equation for $t$ gives $t = 10.94$ years.

Question 10

Write the word or phrase that best completes each statement or answers the question.
-In 1992 , the population of a country was estimated at 5 million. For any subsequent year, the population, $\mathrm { P } ( \mathrm { t } )$ (in millions), can be modeled using the equation $\mathrm { P } ( \mathrm { t } ) = \frac { 250 } { 5 + 44.99 \mathrm { e } ^ { - 0.0208 \mathrm { t } } }$, where $t$ is the number of years since 1992. Determine the year when the population will be 39 million.

Accepted Answer

The answer of Write the word or phrase that best...

Question 11

Choose the one alternative that best completes the statement or answers the question.
-A thermometer reading $84 ^ { \circ } \mathrm { F }$ is placed inside a cold storage room with a constant temperature of $38 ^ { \circ } \mathrm { F }$. If the thermometer reads $77 ^ { \circ } \mathrm { F }$ in 11 minutes, how long before it reaches $56 ^ { \circ } \mathrm { F }$ ? Assume the cooling follows Newton's L of Cooling:
$$\mathrm { U } = \mathrm { T } + \left( \mathrm { U } _ { \mathrm { O } } - \mathrm { T } \right) \mathrm { e } ^ { \mathrm { kt } } .$$
(Round your answer to the nearest whole minute.)
A) $- 20$ minutes
B) 22 minutes
C) 4 minutes
D) 63 minutes

Accepted Answer

The thermometer's temperature change follows an exponential decay model, and reaching $56 ^ { \circ } \mathrm { F }$ from $84 ^ { \circ } \mathrm { F }$ towards $38 ^ { \circ } \mathrm { F }$ would take longer than the initial observed change to $77 ^ { \circ } \mathrm { F }$.

Question 12

Choose the one alternative that best completes the statement or answers the question.
-The function $\mathrm { A } = \mathrm { A } _ { \mathrm { O } } \mathrm { e } ^ { - 0.01386 \mathrm { x } }$ models the amount in pounds of a particular radioactive material stored in a concrete vault, where $x$ is the number of years since the material was put into the vault. If 600 pounds of the material are placed in the vault, how much time will need to pass for only 99 pounds to remain?
A) 140 years
B) 260 years
C) 135 years
D) 130 years

Accepted Answer

The answer of Choose the one alternative that best completes...

Question 13

Choose the one alternative that best completes the statement or answers the question.
-The logistic growth model $\mathrm { P } ( \mathrm { t } ) = \frac { 1,450 } { 1 + 28 \mathrm { e } ^ { - 0.313 \mathrm { t } } }$ represents the population of a bacterium in a culture tube after $\mathrm { t }$ hours. What was the initial amount of bacteria in the population?
A) 55
B) 50
C) 49
D) 51

Accepted Answer

The initial amount of bacteria is found by evaluating the logistic growth model at $ t = 0 $. Substituting $ t = 0 $ into the equation gives $ P(0) = \frac{1450}{1 + 28e^{0}} = \frac{1450}{1 + 28} = \frac{1450}{29} = 50 $.

Question 14

Choose the one alternative that best completes the statement or answers the question.
-Sandy manages a ceramics shop and uses a $650 ^ { \circ } \mathrm { F }$ kiln to fire ceramic greenware. After turning off her kiln, she must wait until its temperature gauge reaches $205 ^ { \circ } \mathrm { F }$ before opening it and removing the ceramic pieces. If room temperature is $70 ^ { \circ } \mathrm { F }$ and the gauge reads $500 ^ { \circ } \mathrm { F }$ in 8 minutes, how long must she wait before opening the kiln? Assume the kiln cools according to Newton's Law of Cooling:
$$\mathrm { U } = \mathrm { T } + \left( \mathrm { U } _ { \mathrm { O } } - \mathrm { T } \right) \mathrm { e } ^ { \mathrm { kt } } .$$
(Round your answer to the nearest whole minute.)
A) 79 minutes
B) 56 minutes
C) 30 minutes
D) 39 minutes

Accepted Answer

Newton's Law of Cooling formula is $U = T + (U_0 - T)e^{kt}$, where $U$ is the temperature of the object at time $t$, $U_0$ is the initial temperature of the object, $T$ is the ambient temperature, and $k$ is a constant. Given that the kiln cools from $650^\circ F$ to $500^\circ F$ in 8 minutes with an ambient temperature of $70^\circ F$, we can solve for $k$ and then use this value to find the time it takes to cool to $205^\circ F$. The calculation involves setting up the equation with the given values and solving for $t$ when $U = 205$. The correct answer, after performing these calculations and rounding to the nearest whole minute, is 39 minutes.

Question 15

Write the word or phrase that best completes each statement or answers the question.
-The temperature (in degrees Fahrenheit) of a dead body that has been cooling in a room set at $70 ^ { \circ }$ is measured as $88 ^ { \circ }$. One hour later, the body temperature is $87.5 ^ { \circ }$. How long (to the nearest hour) before the first measurement was the time of death, assuming that the body temperature of the deceased at the time of death was $98.6 ^ { \circ }$. Assume the cooling follows Newton's Law of Cooling:
$$\mathrm { U } = \mathrm { T } + \left( \mathrm { U } _ { \mathrm { O } } - \mathrm { T } \right) \mathrm { e } ^ { \mathrm { kt } }$$

Accepted Answer

The answer of Write the word or phrase that best...

Question 16

Choose the one alternative that best completes the statement or answers the question.
-The functio models the amount in pounds of a particular radioactive material stored in a $$\mathrm { A } = \mathrm { A } _ { \mathrm { O } } \mathrm { e } ^ { - 0.0099 \mathrm { x } }$$ concrete vault, where x is the number of years since the material was put into the vault. If 500 pounds of the material are initially put into the vault, how many pounds will be left after 80 years?
A) 286 pounds
B) 273 pounds
C) 226 pounds
D) 219 pounds

Accepted Answer

The formula for radioactive decay is given by $ A = A_0 e^{-0.0099x} $. Substituting $ A_0 = 500 $ and $ x = 80 $, we calculate $ A = 500 e^{-0.0099 \times 80} $. This results in approximately 226 pounds.

Question 17

Choose the one alternative that best completes the statement or answers the question.
-A thermometer reading $33 ^ { \circ } \mathrm { F }$ is brought into a room with a constant temperature of $80 ^ { \circ } \mathrm { F }$. If the thermometer reads $45 ^ { \circ } \mathrm { F }$ after 4 minutes, what will it read after being in the room for 6 minutes? Assume the cooling follows Newton's Law of Cooling:
$$\mathrm { U } = \mathrm { T } + \left( \mathrm { U } _ { \mathrm { O } } - \mathrm { T } \right) \mathrm { e } ^ { \mathrm { kt } } \text {. }$$
(Round your answer to two decimal places.)
A) $25.32 ^ { \circ } \mathrm { F }$
B) $71.98 ^ { \circ } \mathrm { F }$
C) $49.8 ^ { \circ } \mathrm { F }$
D) $110.2 ^ { \circ } \mathrm { F }$

Accepted Answer

Using Newton's Law of Cooling, we first find the constant $ k $ using the information that the thermometer reads $45 ^ { \circ } \mathrm { F }$ after 4 minutes. Then, we use this $ k $ to find the temperature after 6 minutes. The calculation shows that the temperature will be approximately $49.8 ^ { \circ } \mathrm { F }$.

Question 18

Choose the one alternative that best completes the statement or answers the question.
-A cup of coffee is heated to $194 ^ { \circ }$ and is then allowed to cool in a room whose air temperature is $72 ^ { \circ }$. After 11 minutes, the temperature of the cup of coffee is $140 ^ { \circ }$. Find the time needed for the coffee to cool to a temperature of $102 ^ { \circ }$. Assume the cooling follows Newton's Law of Cooling:
$$\mathrm { U } = \mathrm { T } + \left( \mathrm { U } _ { \mathrm { O } } - \mathrm { T } \right) \mathrm { e } ^ { \mathrm { kt } } \text {. }$$
(Round your answer to one decimal place.)
A) $15.1$ minutes
B) $29.7$ minutes
C) $41.1$ minutes
D) $26.4$ minutes

Accepted Answer

The answer of Choose the one alternative that best completes...

Question 19

Choose the one alternative that best completes the statement or answers the question.
-The logistic growth model $\mathrm { P } ( \mathrm { t } ) = \frac { 290 } { 1 + 35.25 \mathrm { e } ^ { - 0.188 \mathrm { t } } }$ represents the population of a species introduced into a new territory after t years. What will the population be in 30 years?
A) 290
B) 263
C) 258
D) 2,316

Accepted Answer

The answer of Choose the one alternative that best completes...

Question 20

Choose the one alternative that best completes the statement or answers the question.
-The amount of a certain drug in the bloodstream is modeled by the function $\mathrm { y } = \mathrm { y } _ { 0 } \mathrm { e } ^ { - 0.40 \mathrm { t } }$, where $\mathrm { y } _ { 0 }$ is the amount of the drug injected (in milligrams) and $t$ is the elapsed time (in hours). Suppose that 10 milligrams are injected at 10:00 A.M. If a second injection is to be administered when there is 1 milligram of the drug present in the bloodstream, approximately when should the next dose be given? Express your answer to the nearest quarter hour.
A) 5:45 P.M
B) $5 : 30$ P.M
C) $3 : 45$ P.M
D) $12 : 30$ P.M

Accepted Answer

To find when the next dose should be given, we need to solve for $t$ when $y = 1$ milligram, given that $y_0 = 10$ milligrams. Substituting these values into the equation $y = y_0 e^{-0.40t}$, we get $1 = 10e^{-0.40t}$. Solving for $t$ gives $t \approx 5.75$ hours. Since the initial dose was at 10:00 A.M., adding 5.75 hours leads to approximately 3:45 P.M.

Quiz 4: Exponential and Logarithmic Functions