How long will it take for an investment of 4,000 dollars to double if: a. the interest rate is \( 10 \% \) per year, compounded continuously? (Round your answer to three decimal places) b. the interest rate is \( 8.5 \% \) per year, compounded monthly? (Enter your answer as a whole number of years and months. Round months up to the nearest month where interest is earned. Both blanks must be correct to receive credit.)

We are given an initial investment of $4,000 that is doubled (so the final amount is $8,000). Let P = 4,000 and A = 8,000. The formulas and steps needed for each part are as follows: ───────────────────────────── Part (a): Continuous Compounding For continuous compounding, the formula is:   A = P · e^(rt) When the investment doubles, A/P = 2. So,   2 = e^(rt) Taking the natural logarithm of both sides gives:   ln(2) = rt  ⇒  t = ln(2)/r Here, the annual interest rate is 10% or r = 0.10. Substituting, we get:   t = ln(2)/0.10 Calculating using ln(2) ≈ 0.693147,   t ≈ 0.693147 / 0.10 = 6.93147 years Rounded to three decimal places, t ≈ 6.931 years. ───────────────────────────── Part (b): Monthly Compounding For monthly compounding, the formula is:   A = P · (1 + r/12)^(12t) Again setting A/P = 2 and r = 0.085, we have:   2 = (1 + 0.085/12)^(12t) Taking the natural logarithm of both sides:   ln(2) = 12t · ln(1 + 0.085/12) Solve for t:   t = ln(2) / [12 · ln(1 + 0.085/12)] First calculate the monthly rate:   0.085/12 ≈ 0.00708333 Now compute ln(1 + 0.00708333). Using a calculator,   ln(1.00708333) ≈ 0.00705869 Then,   12 · ln(1.00708333) ≈ 12 × 0.00705869 ≈ 0.08470428 Now,   t ≈ 0.693147 / 0.08470428 ≈ 8.1767 years We now convert the decimal part of the year into months.   Decimal part = 0.1767 years   0.1767 × 12 ≈ 2.1204 months Since you are asked to round the months up to the nearest month where interest is earned, round 2.1204 months up to 3 months. Thus, the investment will approximately double in 8 years and 3 months. ───────────────────────────── Final Answers: a. 6.931 years b. 8 years and 3 months