Yout liave r dollars in Four bank aCcount, which Ol receive A AHAL interest rate of 2%7 COmpounded continously: Write down differential equatiou that describes this...

Money in a bank account earns interest at a continuous annual rate of $5 \%$ times the current balance. Write a differential equation for the balance, $B$, in the account as a function of time, $t,$ in years.

Today we're going to find the effective rate of interest Given that you have 4% compounded continuously in order to find the effective rate of interest. We're going to need to use the formula which is going to be need to the our power minus one which is going to equal your effective rate so we can call that are And so in this question when we're solving for our which is our effective rate, we're going to set up E To the power of our and so instincts are is 4%. We're going to take four. We're going to divide it by a 100 And we're going to get .04 And so we get .04 minus one. So we take our calculator Again, we're going to find our e button which is second Ln You're going to put in your value of .04 And then you are going to close that parentheses and then subtract by one. And so what you would get here for your effective rate is going to be .0408. So if I round that I get zero or which is our effective rate of interest chose that you would get about for christian.

Take and put money into an account, and you have an interest rate of our not gonna continuously. And it wants you to write an equation for how long it takes the money to double. So what we're gonna do is we're gonna write it like this because if you look at a equals P E to the arty, you essentially want a to B to pee. You wanted to double urine stroll or your principal. Okay. And in order to do that, then if you divide by P, you arrive at this two equals either the arty. We want to solve that for tea. It says So what I'm gonna do is I'm gonna take long to then is equal to Artie. And that means that tea is long two divided by our and that's the equation that will tell us how long it will take for something to double

Okay, So what we have here is a bank account that is compounded continuously at an interest rate off 10% or a zero point then and then initial violence in the room. So money is also deposited to the pump account at a continuous rate of $1000 per year. So we are required to get be as a function of f of the Intersil for its solution. So now we can represent being be over BP as point. Then be this 1000. So we want to transfer this so the left side. And we do this by multiplying both sides by one over point. Then be less 1000. Now we'll have baby over point then B plus 1000 bebe physic was one. Now we want to multiply both sides. Baby, be. We'll get Bebe over point, then be this 1000 Is it? Well, so bp so integrating Both sides will have one over 0.1 and and off point. Then be this 1000. Is it? Well, could be less See one. So we multiply both sides by point then and we'll have and then a point. Then be plus 1000 is equal to a point then the plus 0.1 C one. So remember that this is just a constant. So we can therefore 2% this us just see so making both sides of power off e will have books are you You will have a point then be plus 1000 is equal though theories toe point then be let's see So again see is just ah constant value so it can pick up any by use and we can represent our equation. This point then be plus 1000 is equal to C e Raise the point and be if a further isolate b will have B is a well too one over, then one over point then see theories 2.1 B minus. Then thou sent again. This can't just be represented a sea. So our final solution is being is equal to C series the 0.1 b minus then thou son, So we want to find the value of C and we do this by making use off the initial conditions give in from the problem. So at T is equal to zero. B is equal to zero. So substituting this hour, we should will have. Zero is equal to C E. Raise 2.1. Thank zero minus than out, son. So this will just be equal Toe one and solving will have seen is equal to 10,000. So did I think Alverson, who should have be off being is equal to then thou son. Here is the 0.1 B minus, then absent. So this will be our final answer in our part together to new one.

All right. So the question working today is sweet Place. $500. So we placed $500 in a savings account at 5005% interest. Com pounded annually after four years. So it's go. So four years, you withdraw the money and take it to a different bank which advertises a rate of 6%. Compound it annually. That's right. 6% here, 6% competently. What? What is the balance in this new account? After four more years? So the first thing that we're gonna do is for the 1st 4 years we're gonna use the amount equals principle one plus R to the power of end for the formula to solve for this. So this is for the 1st 4 years. So we start with 500 one, plus our interest rate of 0.5 to the power of four. Um, And now when we calculate this and you put in your calculator, you get after the four years we end with $607 and 75.753 So now that's gonna be our starting principle for the next four years. So he's going right next for years, so we're gonna have a is equal to 607.753 one plus and the new interest rate is 6% six 0.6 And we know that's in there for another four years, sort. And it's four. Once again. When you put this into a calculator, you are going to be left with 67 600 or 767 point 274 And that is that's how much balance is going to be in the other bank after the other four years.