SOLUTION: Please help me solve this word problem: A population of grasshoppers quadruples in twenty days. Assuming exponential growth, if the present population is 40 million, what will

Question 1015308: Please help me solve this word problem:
A population of grasshoppers quadruples in twenty days. Assuming exponential growth, if the present population is 40 million, what will it be in fifty 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
Thank-you, your help is always appreciated!
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the population of grasshoppers quadruples in 20 days.

the present population is 40 million.

what will it be in 50 days.

exponential growth formula is f = p * e^(kt).

f is the future population.
p is the present population.
k is the growth per unit time.
t is the amount of time.

the population quadruples every 20 days.

you can solve for k in the following manner.

f = 4
p = 1
t = 20

this formula tells you that your current population is quadrupling in 20 days and it is solving for the growth rate per day.

f = p * e^(kt) becomes 4 = 1 * e^(20k)

this is the same as 4 = e^(20k)

take the natural log of both sides of this equation to get ln(4) = ln(e^(20k))

since ln(e^(20k)) = 20k*ln(e) and since ln(e) = 1, then 20k*ln(e) = 20k.

your formula of ln(4) = ln(e^(20k)) becomes ln(4) = 20k.

divide both sides of this equation by 20 to get ln(4)/20 = k.

solve for k to get k = .0693147181.

k is your daily growth rate.

now that you know the value of k, go back to the original formula of f = p * e^(kt) and replace p with 40 and k with .0693147181 and t with 50 to get:

f = 40 * e^(.0693147181 * 50)

40 is the current population.
.0693147181 is the daily growth rate.
50 is the number of days.

solve for f to get f = 1280 million.

that's the population in 50 days.

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