2. Growth function P=100 (645)^t/21 where P represents the cane toad population and t represents the time measured in months from when the toad was introduced in July 1935.
A. What was the initial population of cane toads introduced into Australia?
B. What was the population after 2 years?
C. What was the date month and year that the population surpassed 1000000 toads?
To find the answers to these questions using the given growth function, we need to substitute the values of t into the equation and perform the necessary calculations.
A. What was the initial population of cane toads introduced into Australia?
To find the initial population, we need to substitute t = 0 into the growth function and solve for P.
P = 100(645)^0/21
P = 100(1)/21
P = 100/21
P ≈ 4.76
Therefore, the initial population of cane toads introduced into Australia was approximately 4.76.
B. What was the population after 2 years?
To find the population after 2 years, we need to substitute t = 24 (since 2 years = 24 months) into the growth function and solve for P.
P = 100(645)^24/21
Calculating this value requires a calculator with a large precision, as the exponent is quite large.
Using a calculator, we find that P ≈ 10,475,142.19.
Therefore, the population after 2 years is approximately 10,475,142.19 toads.
C. What was the date (month and year) that the population surpassed 1,000,000 toads?
To find the date when the population surpassed 1,000,000 toads, we need to set P = 1,000,000 in the growth function and solve for t.
1,000,000 = 100(645)^t/21
To solve this equation, we need to isolate the exponent and use logarithms.
Taking the natural logarithm (ln) of both sides of the equation, we get:
ln(1,000,000) = ln(100(645)^t/21)
ln(1,000,000) = ln(100) + ln((645)^t/21)
Using logarithmic properties, we can simplify further:
ln(1,000,000) = ln(100) + (t/21) ln(645)
Now, we can solve for t by rearranging the equation:
t = (ln(1,000,000) - ln(100))/(ln(645)/21)
Evaluating this expression using a calculator, we find that t ≈ 103.87.
Since t represents the number of months from July 1935, we can calculate the date by adding 103.87 months to July 1935. Taking into account that there are 12 months in a year, we can estimate the date.
Month = July + 7 (months) = February
Year = 1935 + 8 (years) = 1943
Therefore, the estimated month and year when the population surpassed 1,000,000 toads is February 1943.