Question

the growth of a certain species (in millions) since 1960 closely fits the following exponential function where t is the number of years since 1960.$a(t)=3300e^{0.0166t}$a. the population of the species was about 3906 million in 1970. how closely does the function approximate this value?b. use the function to approximate the population of the species in 2000. (the actual population in 2000 was about 6520 million)c. estimate the population of the species in the year 2015.a. choose the correct answer below.a. the population of the species in 1970 is approximately 3896 million. this is close to the actual value of 3906 million, because the percentage difference is very small.b. the population of the species in 1970 is approximately 3896 million. this is not a close approximation to the actual value of 3906 million, because the difference in the two populations is ten million, which is large.c. the population of the species in 1970 is approximately 3396 million. this is not a close approximation to the actual value of 3906 million, because the approximation does not take into account a massive growth increase between 1960 and 1970.b. the approximate population in 2000 is 6410 million.(round to the nearest whole number as needed.)c. the approximate population in 2015 is 8169 million.(round to the nearest whole number as needed.)

Explanation:

Step1: Calculate t for 1970

$t = 1970 - 1960 = 10$

Step2: Compute A(10)

$A(10) = 3300e^{0.0166 \times 10} = 3300e^{0.166} \approx 3896$

Step3: Compare to actual 1970 value

Percent difference: $\frac{|3896 - 3906|}{3906} \times 100\% \approx 0.26\%$, very small.

Step4: Calculate t for 2000

$t = 2000 - 1960 = 40$

Step5: Compute A(40)

$A(40) = 3300e^{0.0166 \times 40} = 3300e^{0.664} \approx 6410$

Step6: Calculate t for 2015

$t = 2015 - 1960 = 55$

Step7: Compute A(55)

$A(55) = 3300e^{0.0166 \times 55} = 3300e^{0.913} \approx 8169$

Answer:

a. A. The population of the species in 1970 is approximately 3896 million. This is close to the actual value of 3906 million, because the percentage difference is very small.
b. 6410
c. 8169