Question 1

Provide an appropriate response.
-In 2000, the population of Littletown was 16 thousand. Use the given doubling time to predict the population in 2060. Assume a doubling time of 30 years.

Accepted Answer

The population doubles every 30 years. From 2000 to 2060 is 60 years, meaning the population will double twice. Starting with 16 thousand, it doubles to 32 thousand in 2030, and then doubles again to 64 thousand by 2060.

Question 2

Provide an appropriate response.
-Given $\log _ { 10 } 2 = 0.3010$ and $\log _ { 10 } 3 = 0.4771$, find $\log _ { 10 } 12$ without using a calculator.

Accepted Answer

Using the given logarithm values, $\log _ { 10 } 12$ can be found by expressing 12 as $2 \times 2 \times 3$ or $2^2 \times 3$. Thus, $\log _ { 10 } 12 = \log _ { 10 } (2^2 \times 3) = 2\log _ { 10 } 2 + \log _ { 10 } 3 = 2(0.3010) + 0.4771 = 0.6020 + 0.4771 = 1.0791$.

Question 3

Use the decibel scale to answer the question.
-How does the intensity of sound form a concert speaker at a distance of 1 meter compare to the intensity at a distance of 800 meters?

Accepted Answer

The intensity of sound decreases with the square of the distance from the source, according to the inverse square law. Therefore, if the distance increases by a factor of 800 (from 1 meter to 800 meters), the intensity decreases by a factor of $800^2 = 640,000$, making choice B correct.

Question 4

Provide an appropriate response.
-The current population of a threatened animal species is 1 million, but it is declining with a half-life of 27 years. How many animals will be left in 74 years?

Accepted Answer

To find the remaining population after 74 years, we use the half-life formula. With a half-life of 27 years, after 74 years (approximately 2.74 half-lives), the population would be $1,000,000 \times (1/2)^{2.74}$, which is approximately 149,608 animals.

Question 5

Solve.
-Use the magic penny parable to determine how many days would elapse before you had a total of over 155,235.

Accepted Answer

The answer of Solve.
-Use the magic penny parable to determine...

Question 6

Provide an appropriate response.
-What are the conditions in which the approximate doubling time formula works well?

Accepted Answer

The approximate doubling time formula, often referred to as the Rule of 70, works well for small growth rates and tends to break down or become less accurate for growth rates over 15%.

Question 7

Use the decibel scale to answer the question.
-How does the intensity of sound from a concert speaker at a distance of 20 meters compare to the intensity at a distance of 140 meters?

Accepted Answer

The intensity of sound decreases with the square of the distance from the source, so if the distance increases by a factor of 7 (from 20 m to 140 m), the intensity decreases by a factor of 7^2, which is 49.

Question 8

Provide an appropriate response.
-Use the 1960s peak annual growth rate of 2.5% and population of 4 million to predict the current growth rate with a logistic model. Assume a current country population of 10 million. Assume the
Carrying capacity is 21 million.

Accepted Answer

The logistic growth model formula is r = r_max * (1 - P/K), where r_max is the peak growth rate, P is the current population, and K is the carrying capacity. Substituting the given values: r = 2.5% * (1 - 10/21) = 1.62%.

Question 9

Provide an appropriate response.
-Poaching is causing a population of elephants to decline by 6% per year. Use the approximate half-life formula to determine the fraction that remains in 76 years if there are 13,365 elephants
Today.

Accepted Answer

The answer of Provide an appropriate response.
-Poaching is causing a...

Question 10

Provide an appropriate response.
-Inflation is causing prices to rise at a rate of 6% per year. Use the exact double time formula to determine what the price will be in 4 years if the item costs $100 today.

Accepted Answer

The exact double time formula isn't directly applicable here, but the general formula for future value in the context of inflation is $FV = PV(1 + r)^n$, where $PV$ is the present value, $r$ is the annual rate (expressed as a decimal), and $n$ is the number of years. For a $100 item with a 6% inflation rate over 4 years, the calculation is $100(1 + 0.06)^4 = 100(1.2625) = $126.25$.

Question 11

Provide an appropriate response.
-The half-life of a radioactive substance is 65 years. If you start with some amount of this substance, what fraction will remain in 132 years?

Accepted Answer

In 132 years, the substance will go through 132 / 65 = 2.03 half-lives, approximately 2 half-lives. After one half-life, half of the substance remains, and after two half-lives, half of that amount remains, so 1/2 * 1/2 = 1/4 or 0.25. The answer closest to this calculation is 0.24.

Question 12

Provide an appropriate response.
-The initial population of a town is 128,066 and it grows with a doubling time of 97 years. What will the population be in 35 years?

Accepted Answer

The population growth can be calculated using the formula for exponential growth: $P = P_0 \times (2)^{\frac{t}{T}}$, where $P$ is the future population, $P_0$ is the initial population, $t$ is the time in years, and $T$ is the doubling time in years. Plugging in the given values: $P = 128,066 \times (2)^{\frac{35}{97}}$, which calculates to approximately 164,457 people.

Question 13

Provide an appropriate response.
-The number of cells in a tumor doubles every 3.7 months. If the tumor begins as a single cell, how many cells will there be after 3 years?

Accepted Answer

The number of times the cells double in 3 years (36 months) is 36 / 3.7 ≈ 9.73 times. Since cells can only double a whole number of times, we consider 9 full doublings. Starting with one cell, after 9 doublings (2^9), there will be 512 cells. However, the closest answer provided is 849 cells, indicating a possible miscalculation in the interpretation of the doubling period or the options provided. The correct calculation method involves using the formula for exponential growth, but none of the provided options accurately reflects the exact number of cells after 3 years based on the given doubling rate. The explanation aims to follow the logic of doubling, but the discrepancy suggests a misunderstanding or a typo in the options.

Question 14

Provide an appropriate response.
-A certain radioactive isotope decays at a rate of 0.15% per year. Determine the half-life of this isotope, to the nearest year.

Accepted Answer

The half-life of a radioactive isotope can be calculated using the formula $T_{1/2} = \frac{\ln(2)}{r}$, where $T_{1/2}$ is the half-life and $r$ is the decay rate per period. Given a decay rate of 0.15% per year, or $r = 0.0015$ per year, the half-life is $T_{1/2} = \frac{\ln(2)}{0.0015} \approx 462$ years.

Question 15

Use the decibel scale to answer the question.
-What is the loudness, in decibels, of a sound 16 trillion times as loud as the softest audible sound?

Accepted Answer

The answer of Use the decibel scale to answer the...

Question 16

Provide an appropriate response.
-Use the 1960s peak annual growth rate of 0.4% and population of 6 million to predict the current growth rate with a logistic model. Assume a current country population of 15 million. Assume the
Carrying capacity is 26 million.

Accepted Answer

The logistic growth rate can be calculated using the formula $ r = r_0 \left(1 - \frac{P}{K}\right) $, where $r_0$ is the initial growth rate (0.4%), $P$ is the current population (15 million), and $K$ is the carrying capacity (26 million). Plugging in the numbers: $ r = 0.4\% \left(1 - \frac{15}{26}\right) \approx 0.22\% $.

Question 17

Provide an appropriate response.
-Oil consumption is increasing at a rate of 2.1% per year. By what factor will oil consumption increase in 2 years? Use the approximate doubling time formula (rule of 70).

Accepted Answer

The rule of 70 is used to estimate the doubling time of an investment or in this case, the growth of oil consumption. However, to find the factor by which oil consumption increases in 2 years at a rate of 2.1% per year, we use the formula for compound growth: $F = P(1 + r)^t$, where $F$ is the future value, $P$ is the present value (which can be taken as 1 for simplicity), $r$ is the rate of increase (2.1% or 0.021), and $t$ is the time in years (2 years in this case). Plugging in the values, we get $F = 1(1 + 0.021)^2$, which simplifies to approximately 1.04. This means the oil consumption will increase by a factor of 1.04 in 2 years, not double. The rule of 70 is not directly used for this calculation; it's more for estimating doubling time, not the factor of increase over a specific period.

Question 18

Use the given growth rate to find the approximate doubling time and to predict the population in 50 years of a growingsuburban town (based on a current population of 100,000).
-Use the current annual growth rate which is about 3.1%.

Accepted Answer

The doubling time can be estimated using the Rule of 70: $ \frac{70}{3.1} \approx 22.58 $ years, which rounds to 23 years. Using the formula $ P = P_0(1 + r)^t $, where $ P_0 = 100,000 $, $ r = 0.031 $, and $ t = 50 $, the population in 50 years is approximately 464,052.

Question 19

Provide an appropriate response.
-The carrying capacity of our planet depends on our consumption of energy.

Accepted Answer

The carrying capacity of our planet, or the maximum population size that the environment can sustain indefinitely, is influenced by our consumption of energy because energy consumption affects resource use, pollution levels, and ecological footprints, all of which impact the planet's ability to support human life.

Question 20

Use the pH scale to answer the question.
-What is the pH of a solution with a hydrogen ion concentration of 0.001 mole per liter? Is this solution an acid or a base?

Accepted Answer

The pH is calculated using the formula $\mathrm{pH} = -\log[H^+]$, where $[H^+]$ is the hydrogen ion concentration. For a concentration of 0.001 mole per liter, $\mathrm{pH} = -\log(0.001) = 3$. A pH less than 7 indicates an acidic solution.

Quiz 8: Exponential Astonishment

Provide an appropriate response. -In 2000, the population of Littletown was 16 thousand. Use the given doubling time to predict the population in 2060. Assume a doubling time of 30 years.

Provide an appropriate response. -Given $\log _ { 10 } 2 = 0.3010$ and $\log _ { 10 } 3 = 0.4771$ , find $\log _ { 10 } 12$ without using a calculator.

Use the decibel scale to answer the question. -How does the intensity of sound form a concert speaker at a distance of 1 meter compare to the intensity at a distance of 800 meters?

Provide an appropriate response. -The current population of a threatened animal species is 1 million, but it is declining with a half-life of 27 years. How many animals will be left in 74 years?

Solve. -Use the magic penny parable to determine how many days would elapse before you had a total of over 155,235.

Provide an appropriate response. -What are the conditions in which the approximate doubling time formula works well?

Use the decibel scale to answer the question. -How does the intensity of sound from a concert speaker at a distance of 20 meters compare to the intensity at a distance of 140 meters?

Provide an appropriate response. -Use the 1960s peak annual growth rate of 2.5% and population of 4 million to predict the current growth rate with a logistic model. Assume a current country population of 10 million. Assume the Carrying capacity is 21 million.

Provide an appropriate response. -Poaching is causing a population of elephants to decline by 6% per year. Use the approximate half-life formula to determine the fraction that remains in 76 years if there are 13,365 elephants Today.

Provide an appropriate response. -Inflation is causing prices to rise at a rate of 6% per year. Use the exact double time formula to determine what the price will be in 4 years if the item costs $100 today.

Provide an appropriate response. -The half-life of a radioactive substance is 65 years. If you start with some amount of this substance, what fraction will remain in 132 years?

Provide an appropriate response. -The initial population of a town is 128,066 and it grows with a doubling time of 97 years. What will the population be in 35 years?

Provide an appropriate response. -The number of cells in a tumor doubles every 3.7 months. If the tumor begins as a single cell, how many cells will there be after 3 years?

Provide an appropriate response. -A certain radioactive isotope decays at a rate of 0.15% per year. Determine the half-life of this isotope, to the nearest year.

Use the decibel scale to answer the question. -What is the loudness, in decibels, of a sound 16 trillion times as loud as the softest audible sound?

Provide an appropriate response. -Use the 1960s peak annual growth rate of 0.4% and population of 6 million to predict the current growth rate with a logistic model. Assume a current country population of 15 million. Assume the Carrying capacity is 26 million.

Provide an appropriate response. -Oil consumption is increasing at a rate of 2.1% per year. By what factor will oil consumption increase in 2 years? Use the approximate doubling time formula (rule of 70) .

Use the given growth rate to find the approximate doubling time and to predict the population in 50 years of a growingsuburban town (based on a current population of 100,000) . -Use the current annual growth rate which is about 3.1%.

Provide an appropriate response. -The carrying capacity of our planet depends on our consumption of energy.

Use the pH scale to answer the question. -What is the pH of a solution with a hydrogen ion concentration of 0.001 mole per liter? Is this solution an acid or a base?