calculus question

Question

A population of grasshoppers quadruples in twenty days. Assuming exponential growth, if the present population is 40 million, what will it be in 50 days? Answer the question by first finding the number y of grasshoppers as a function of time t (in days) in the form y = y₀e^kt. 

 

Help!?

Answer 1

A way to solve this by going directly to the form y = y_oe^kt, with t in days:

 

At t = 0, you have y₀.  After 20 days, you have 4y₀.

 

4y₀ = y₀e^k(20)

 

4 = e^20k

 

k = (ln4) / 20 ≅ 0.06931

 

y(t) = (40 million) e^0.06931t

 

y(50) = (40 million) e^0.06931(50) ≅ 1279.7 million

 

 

Answer 2

y = y₀·4^t/20

• y = number of grasshoppers at time t
• y₀ = initial number of grasshoppers = 40 million
• 4 = rate of change = quadruples = 4

To put it into the form y = y₀e^kt, note that 4 = e^ln(4) and ln(a^b) = b·ln(a) so:

 

     y = 40·4^t/20 =40·e^{ln(4^t/20)} =40 ·e^{t/20 ·ln(4)} = 40·e^ln(4)/20·t 

 

Now k = ln(4)/20 ≅ 0.069, so the equation is:

 

     y = 40·e^0.069·t

 

To find the population in t = 50 days:

 

     y = 40·e^0.069·t

     y = 40·e^0.069·50

 

Use your calculator to get the answer.  The answer is in millions.  It's a lot of grasshoppers.