The Population Of A Bee Colony Is Growing At 23 Each
The Population Of A Bee Colony Is Growing At A Rate Of 23 Each Y
The population of a bee colony is growing at a rate of 2.3% each year. There are currently 3400 bees in the colony. At this rate, in how many years will there be 10,200 bees in the colony?
The population of Mar Mac is increasing by 6% each year. How many years will it take for the population to double?
How many years will it take for the population to triple?
The population of fish in a lake is decreasing. There are currently 24,000 fish in the lake. The population is decreasing by 5.75% each year. In how many years will the lake only contain a fourth of its current population of fish?
The half-life of Radium is 1690 years. If 10 grams of Radium are present now, how much Radium will be present in 50 years?
Suppose the half-life of ibuprofen in the bloodstream is 45 minutes. If you take 400 mg of ibuprofen, how much will remain in your bloodstream in 5 hours?
Suppose a material decays at a rate proportional to the quantity of the material and there were 2500 grams of the material 10 years ago. If there are 2400 grams of the material now, what is the half-life of the material?
Paper For Above instruction
Exponential growth and decay models are fundamental in understanding various real-world phenomena, such as population dynamics, radioactive decay, and pharmacokinetics. These models are expressed mathematically as P(t) = P_0 e^{rt} for growth, where P_0 is the initial population, r is the growth rate, and t is time, and as P(t) = P_0 e^{rt} with a negative r for decay. This paper explores several scenarios involving exponential models, including population growth, decline, and radioactive decay, providing mathematical solutions and interpretations for each case.
Population Growth of a Bee Colony
The initial population is 3400 bees, and the growth rate is 2.3% per year, which translates to a growth factor of 1 + 0.023 = 1.023. The problem asks when the population will reach 10,200 bees. Using the exponential growth formula:
P(t) = P_0 e^{rt}
We can also use the equivalent form P(t) = P_0 (1 + r)^t for discrete annual growth, thus:
10,200 = 3400 * (1.023)^t
Dividing both sides by 3400 gives:
(10,200 / 3400) = (1.023)^t
Which simplifies to:
3 = (1.023)^t
Taking natural logarithms of both sides:
ln(3) = t * ln(1.023)
Therefore:
t = ln(3) / ln(1.023) ≈ 1.0986 / 0.02275 ≈ 48.3 years
It will take approximately 48.3 years for the bee population to grow from 3400 to 10200 bees at this rate.
Population Doubling and Tripling
For Mar Mac's population increasing by 6% annually, the growth factor is 1.06. To find the time to double the population:
2 = (1.06)^t
ln(2) = t * ln(1.06)
t = ln(2) / ln(1.06) ≈ 0.6931 / 0.05827 ≈ 11.9 years
Similarly, to triple the population:
3 = (1.06)^t
t = ln(3) / ln(1.06) ≈ 1.0986 / 0.05827 ≈ 18.86 years
Thus, it takes approximately 11.9 years to double and 18.86 years to triple the population at this rate.
Fish Population Decrease in a Lake
The current fish population is 24,000, decreasing at 5.75% annually, which corresponds to a decay factor of 1 - 0.0575 = 0.9425. We seek the time when the population reduces to one-fourth of its current size, i.e., 6000 fish.
Using the decay formula:
6000 = 24,000 * (0.9425)^t
Dividing both sides by 24,000:
0.25 = (0.9425)^t
Taking natural logarithms:
ln(0.25) = t * ln(0.9425)
t = ln(0.25) / ln(0.9425) ≈ -1.3863 / -0.05946 ≈ 23.3 years
It will take approximately 23.3 years for the fish population to reduce to a quarter.
Radioactive Decay of Radium
Radium's half-life is 1690 years, meaning every 1690 years, half of the substance decays. Starting with 10 grams, the amount remaining after t years is:
A(t) = A_0 * (1/2)^{t / T_{1/2}}
Substituting the known values:
A(50) = 10 (1/2)^{50 / 1690} ≈ 10 (1/2)^{0.0296} ≈ 10 * 0.979 ≈ 9.79 grams
So, after 50 years, about 9.79 grams of Radium remain.
Decay of Ibuprofen in the Human Body
Ibuprofen has a half-life of 45 minutes. The exponential decay formula applies:
A(t) = A_0 * (1/2)^{t / T_{1/2}}
Initial dose A_0 = 400 mg, time t = 5 hours = 300 minutes:
A(300) = 400 (1/2)^{300 / 45} = 400 (1/2)^{6.6667} ≈ 400 * 0.0104 ≈ 4.16 mg
Thus, approximately 4.16 mg of ibuprofen remains in the bloodstream after 5 hours.
Decay Rate and Half-Life of a Material
Given that 2500 grams of a material was present 10 years ago, and now (at t=0) there are 2400 grams, the decay rate is very slow, and the model follows:
P(t) = P_0 * e^{kt}
Using data:
2400 = 2500 e^{k 10}
e^{10k} = 2400/2500 = 0.96
10k = ln(0.96) ≈ -0.0408
k ≈ -0.00408
The half-life T_{1/2} is given by:
T_{1/2} = ln(2) / |k| ≈ 0.6931 / 0.00408 ≈ 169.8 years
The material's half-life is approximately 170 years, similar to Radium, indicating slow decay.
Conclusion
These calculations demonstrate the application of exponential models to real phenomena. Growth models enable predictions of population increases over time, while decay models are crucial in radioactive decay, pharmacokinetics, and understanding the longevity of materials. Accurate modeling relies on understanding the underlying principles of exponential functions, logarithms, and half-life concepts. Such models are invaluable in fields like ecology, medicine, geology, and physics, providing critical insights into processes occurring over varying timescales.
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