SOLUTION: PLEASE PLEASE. I'M IN DESPERATE NEED OF HELP! A mosquito population is growing at a rate of 8% each day and the current population is estimated at 200,000. Use y=y sub 0 times e

Question 84669: PLEASE PLEASE. I'M IN DESPERATE NEED OF HELP!
A mosquito population is growing at a rate of 8% each day and the current population is estimated at 200,000. Use y=y sub 0 times e^0.08t as the equation for the mosquito population.
a) how many mosquitoes are there expected to be in 10 days?
b) how many days will it take for the mosquito population to reach one million?

Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Given:
.

.
At time t = 0, the mosquito population represented in this equation by y is 200,000.
.
t represents the number of days since day zero when t = 0
.
Note that on day zero (when t = 0) the part of the equation represented by:
.

.
becomes
.
because any number to the 0 power is 1 by definition.
.
Since this is 1 the equation is reduced to and since the mosquito population
is known to be 200,000 at this time, we know that . So we can substitute
200,000 for and the equation becomes:
.

.
Now to the problem. In ten days (when t = 10) the population of mosquitoes becomes:
.

.
Calculator time. . Substitute this into the equation and it becomes:
.

.
Rounding to the nearest whole number, the equation tells us that in 10 days there will
be 445108 mosquitoes.
To find out how many days it will take for the mosquito population to reach a million, set
y equal to 1,000,000 in the equation and solve for t. The equation becomes:
.

.
Simplify by dividing both sides by 200,000 to get:
.

.
Take the natural log of both sides and you get:
.

.
The ln of 5 is (by calculator) 1.609437912. Substitute this:
.

.
And by the power rule of logarithms, in the log of a quantity that is raised to an exponent,
you can bring the exponent out and make it the multiplier of the log of the quantity that
remains. In this case that means . Substitute the right
side into the equation in place of and the result is:
.

.
But ln(e) = 1. Substitute this and the equation reduces to:
.

.
Solve for t by dividing both sides of this equation by 0.08 and the answer becomes:
.

.
So it will take a little more than 20 days for the colony of mosquitoes to be more than
one million in population. The population should slightly exceed a million in 20 days and
3 hours.
.
Hope this helps you to understand the problem.
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