Exponential growth

Question

A skull cleaning factory cleans animal skulls of deer, buffalo, and other types of animals using flesh-eating beetles. The factory owner started with only 10 adult beetles. After 34 days, the beetle population grew to 40 beetles. Assuming uninhibited exponential growth, how long did it take before the beetle population was 10,000 beetles?

It took (blank) days.

Answer 1

Hi Lexes..

A skull cleaning factory cleans animal skulls of deer, buffalo, and other types of animals using flesh-eating beetles. The factory owner started with only 10 adult beetles. After 34 days, the beetle population grew to 40 beetles. Assuming uninhibited exponential growth, how long did it take before the beetle population was 10,000 beetles?

Given:

Started with 10 adult beetles

after 34 days had 40 beetles

assume uninhibited exponential growth

Find:

The time to get to 10,000 beetles

Exponential Growth Equation

y=y₀e^kt

Where y₀= initial conditions, k=rate of growth, t= time

y₀= 10 This is the original amount of beetles

We need to determine the value of k.

Using the data point given (40 beetles after 34 days) we can calculate k

y=y₀e^kt

40 = 10e^k(34) / Note the unit for t is days since we were given the info in days

4=e^34k

ln 4 = 34k

k = 0.04077

So our exponential equation is

y=10e^(0.04077)t

Now that we have the value for k we can determine the time it would take to get to 10k beetles

y=10e^(0.04077)t

10,000 = 10 e^(0.04077)t

t= 169.4 days

Verify.. Plug in 169.4 for t and make sure you get the 10k you expected

y=10e^(0.04077)t

y=10e^{(0.04077)(169.4)}

y = 9986.8 so approximately 10k

Hope this helps :)