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5200 dollars is placed in an account with an annual interest rate of 5.75%. To the nearest year, how long will it take for the account value to reach 22500 dollars? Answer:

52005200 dollars is placed in an account with an annual interest rate of 5.75% 5.75 \% . To the nearest year, how long will it take for the account value to reach 2250022500 dollars?\newlineAnswer:

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Q. 52005200 dollars is placed in an account with an annual interest rate of 5.75% 5.75 \% . To the nearest year, how long will it take for the account value to reach 2250022500 dollars?\newlineAnswer:
  1. Identify Interest Type: Determine the type of interest being applied.\newlineSince the problem does not specify compound or simple interest, we will assume compound interest is applied annually.
  2. Compound Interest Formula: Identify the formula for compound interest.\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.\newlineSince interest is compounded annually, n=1n = 1.
  3. Convert Interest Rate: Convert the annual interest rate from a percentage to a decimal.\newliner=5.75%=0.0575r = 5.75\% = 0.0575
  4. Set Up and Solve Equation: Set up the equation with the given values and solve for tt. We have P=5200P = 5200, A=22500A = 22500, r=0.0575r = 0.0575, and n=1n = 1. 22500=5200(1+0.0575/1)(1⋅t)22500 = 5200(1 + 0.0575/1)^{(1\cdot t)}
  5. Simplify and Solve for tt: Simplify the equation and solve for tt.22500=5200(1+0.0575)t22500 = 5200(1 + 0.0575)^tDivide both sides by 52005200 to isolate the exponential part of the equation.225005200=(1+0.0575)t\frac{22500}{5200} = (1 + 0.0575)^t4.32692307692≈(1.0575)t4.32692307692 \approx (1.0575)^t
  6. Use Logarithms to Solve: Use logarithms to solve for tt. Take the natural logarithm (ln) of both sides to get rid of the exponent. ln⁡(4.32692307692)=ln⁡((1.0575)t)\ln(4.32692307692) = \ln((1.0575)^t) Use the power rule of logarithms: ln⁡(ab)=b⋅ln⁡(a)\ln(a^b) = b\cdot\ln(a). ln⁡(4.32692307692)=t⋅ln⁡(1.0575)\ln(4.32692307692) = t \cdot \ln(1.0575)
  7. Calculate t Value: Calculate the value of tt.t=ln⁡(4.32692307692)ln⁡(1.0575)t = \frac{\ln(4.32692307692)}{\ln(1.0575)}t≈ln⁡(4.32692307692)ln⁡(1.0575)t \approx \frac{\ln(4.32692307692)}{\ln(1.0575)}t≈1.465103131570.055892553t \approx \frac{1.46510313157}{0.055892553}t≈26.21t \approx 26.21
  8. Round to Nearest Year: Round the value of tt to the nearest year.\newlineSince we need to find the nearest year, we round 26.2126.21 to 2626 years.

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