Suppose that $P$ dollars in principal is invested in an account earning $3.5 \%$ interest compounded continuously. At the end of $4 \mathrm{yr}$, the amount in the account has earned $\$ 976.78$ in interest.
Part: $0 / 2$
Part 1 of 2
(a) Find the original principal, Round to the nearest dollar. (Hint: Use the model $A=P e^{r t}$ and substitute $P+97678$ for $A$.)
The original principal was approximately $\$ \square$.
Answer
Rounding to the nearest dollar gives the original principal as approximately \(\boxed{6500}\) dollars.
Step 1 ：Given that the final amount in the account is the original principal plus the interest earned, which is \(P + 976.78\). The interest rate is \(3.5\% = 0.035\) and the time is \(4\) years.
Step 2 ：Substitute these values into the formula for continuous compound interest, \(A = P e^{rt}\), and solve for \(P\).
Step 3 ：Solving the equation gives \(P\) approximately equal to 6500.00204578594.
Step 4 ：Rounding to the nearest dollar gives the original principal as approximately \(\boxed{6500}\) dollars.
