# Math Help - Differential Equation

1. ## Differential Equation

Sorry about the length, and if someone could do at least one of these, I think I'll be able to recall the method of setting these up to solve the others. I just cannot remember how to do these types of problems though.

If someone could work it, or a similar problem, step-by-step so I can see the method, I would greatly appreciate it.

1) A student borrows \$8000 to buy a car. The lender charges interest at an annual rate of 10%. Assuming that interest is compounded continuously and that the borrower makes payments continuously at a constant annual rate k, determine the payment rate k that is required to pay off the loan in 3 years. Also determine how much interest is paid during the three year period.

2) Consider a tank used in certain hydrodynamic experiments. After one experiment, the tank contains 200 L of a dye solution with concentration of 1 g/L. To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of 2 L/min, the well-stirred solution flowing out at the same rate. Find the time that will elapse before the concentration of dye in the tank reaches 1% of its original value.

3) The population of mosquitoes in a certain area increases at a rate proportional to the current population, and, in the absence of other factors, the population doubles each week. There are 200,000 mosquitoes in the area initially, and predators eat 20,000 mosquitoes/day. Determine the population of mosquitoes in the area at any time.

2. Originally Posted by Math Major
2) Consider a tank used in certain hydrodynamic experiments. After one experiment, the tank contains 200 L of a dye solution with concentration of 1 g/L. To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of 2 L/min, the well-stirred solution flowing out at the same rate. Find the time that will elapse before the concentration of dye in the tank reaches 1% of its original value.
The General idea is this...

$\frac{dA}{dt}=r_i-r_o$ where A is the amount in the tank

$r_i$ is the rate in and $r_o$ is the rate out

Since the tank is 200L and the concentration is 1g/L we have 200g at time t=0.

The rate of die comming in is zero.(They are only adding pure water)

$r_0=\left( \frac{A}{200L}\right) \left( \frac{2L}{1 min}\right)=\frac{A}{100}$

So we get the ODE

$\frac{dA}{dt}=0-\frac{A}{100}$ with intial condition

$A(0)=0$

This is first order linear and seperable. So solving we get

$A(t)=200e^{-\frac{t}{100}}$

Good luck.

TES