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8700 dollars is placed in an account with an annual interest rate of 8%. To the nearest year, how long will it take for the account value to reach 20000 dollars? Answer:

87008700 dollars is placed in an account with an annual interest rate of 8% 8 \% . To the nearest year, how long will it take for the account value to reach 2000020000 dollars?\newlineAnswer:

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Q. 87008700 dollars is placed in an account with an annual interest rate of 8% 8 \% . To the nearest year, how long will it take for the account value to reach 2000020000 dollars?\newlineAnswer:
  1. Identify Type of Interest: Identify the type of interest being applied.\newlineSince the problem does not specify compound or simple interest, we will assume compound interest is being applied annually.
  2. Use Compound Interest Formula: Use the formula for compound interest to find the time.\newlineThe formula for compound interest is A=P(1+r/n)(nt)A = P(1 + r/n)^{(nt)}, where:\newlineAA = the amount of money accumulated after nn years, including interest.\newlinePP = the principal amount (the initial amount of money).\newlinerr = the annual interest rate (decimal).\newlinenn = the number of times that interest is compounded per year.\newlinett = the time the money is invested for, in years.\newlineIn this case, A=$20000A = \$20000, P=$8700P = \$8700, r=8%r = 8\% or AA00, and AA11 (since it's compounded annually).
  3. Convert Rate to Decimal: Convert the percentage rate to a decimal and plug in the values.\newlineA=20000A = 20000, P=8700P = 8700, r=0.08r = 0.08, n=1n = 1.\newlineWe need to solve for tt.\newline20000=8700(1+0.08/1)(1⋅t)20000 = 8700(1 + 0.08/1)^{(1\cdot t)}
  4. Simplify Equation and Solve: Simplify the equation and solve for tt.20000=8700(1+0.08)t20000 = 8700(1 + 0.08)^t20000=8700(1.08)t20000 = 8700(1.08)^t
  5. Isolate Exponential Part: Divide both sides by 87008700 to isolate the exponential part.\newline200008700=(1.08)t \frac{20000}{8700} = (1.08)^t \newline2.29885057471≈(1.08)t2.29885057471 \approx (1.08)^t
  6. Take Natural Logarithm: Take the natural logarithm of both sides to solve for tt.ln⁡(2.29885057471)=ln⁡((1.08)t)\ln(2.29885057471) = \ln((1.08)^t)ln⁡(2.29885057471)=t⋅ln⁡(1.08)\ln(2.29885057471) = t \cdot \ln(1.08)
  7. Divide to Solve for t: Divide both sides by ln⁡(1.08)\ln(1.08) to solve for t.t=ln⁡(2.29885057471)ln⁡(1.08)t = \frac{\ln(2.29885057471)}{\ln(1.08)}t≈10.2447683516t \approx 10.2447683516
  8. Round to Nearest Year: Round the answer to the nearest year.\newlinet≈10t \approx 10 years

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