Identifying the values of the variables in the formula.
Determining which variable we need to solve for.
Determining Formula
We have to use the formula below because the annual interest is compounded continuously.
A( t)= P * e^(r t)
In this formula, A( t) is the amount in the account in time t, P is the principal, r is the annual interest rate.
Identifying the Values
We know that an account pays 6.5 % annual interest rate compounded continuously. We want to find the number of years necessary for our principal to double. Let say the principal is equal to P. After t years, it will double itself, 2P.
Therefore,
A( t)= 2P,
P= P
r=6.5 %=0.065,
Determining the Variable to Solve
We need to find the number of years t for our principal to double.
Let's substitute the values into the formula.
To solve the equation by graphing, we first have to create two functions. Each side of the given equation will become its own equation.
2= e^(0.065 * t) ⇒ ly=e^(0.065 * t) y=2
Now we want to draw these graphs using a calculator. We can do this by pressing the Y= button and typing the equations in the two first rows. We replace the variable t with the variable x.
Before we graph these functions, notice that the exponent of e is 0.065x, so the function will grow very slowly. We may not be able to see the intersection of the lines in the standard viewing window. That's why, we should change the viewing window. We can do this by pushing WINDOW.
There is one point of intersection of these lines. To find this point we can use the intersect option, which we get by pushing 2nd and TRACE. Now, we select both graphs and provide the calculator with a guess as to where the intersection might be.
The solution to the equation is x ≈ 10.66. It would take approximately 10.66 years to double our principal.
