Final answer:
To calculate the number of years for an investment to double at a 10% continuously compounded rate of interest, the formula Pt=P0ert is used with Pt being twice P0. After simplifying, solving for t entails the calculation t = ln(2) / 0.10, which is approximately 6.93, thus t is approximately 7 years.
Step-by-step explanation:
To find out the time t it takes for an investment to double when earning interest at a continuous rate, we use the given formula Pt=P0ert, where Pt is the amount after t years, P0 is the initial amount, r is the rate of interest, and t is the time in years. To double the investment, Pt becomes 2P0. Thus, the equation becomes 2P0=P0ert. After simplifying and dividing both sides by P0, we get 2=ert. Taking the natural logarithm of both sides, we have ln(2)=rt. Since the rate r is 10% (or 0.10 as a decimal), we substitute this into our equation, giving us ln(2)=0.10t. Solving for t, we get t = ln(2) / 0.10.
We can use a calculator to find ln(2) ≈ 0.693. Dividing this by 0.10, we get t ≈ 6.93, which we round up to 7 years since the exact value is not one of the given options. Therefore, we can conclude that the correct answer is t=7 years.