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A house purchased 5 years ago for $100,000 was just sold for $135,000. Assuming exponential growth, approximate the annual growth rate, to the nearest percent. Work/Explanation

A house purchased 55 years ago for $100,000 \$ 100,000 was just sold for $135,000 \$ 135,000 . Assuming exponential growth, approximate the annual growth rate, to the nearest percent.\newlineWork/Explanation

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Q. A house purchased 55 years ago for $100,000 \$ 100,000 was just sold for $135,000 \$ 135,000 . Assuming exponential growth, approximate the annual growth rate, to the nearest percent.\newlineWork/Explanation
  1. Set Up Equation: To find the annual growth rate, we will use the formula for exponential growth, which is:\newlineFinal Value = Initial Value ×(1+r)t\times (1 + r)^t\newlinewhere:\newlineFinal Value = $135,000\$135,000\newlineInitial Value = $100,000\$100,000\newliner = annual growth rate (as a decimal)\newlinet = time in years, which is 55 years in this case\newlineWe need to solve for rr.
  2. Isolate Growth Factor: First, we divide both sides of the equation by the Initial Value to isolate the growth factor on one side:\newline$135,000$100,000=(1+r)5\frac{\$135,000}{\$100,000} = (1 + r)^5\newline1.35=(1+r)51.35 = (1 + r)^5
  3. Calculate Fifth Root: Next, we take the fifth root of both sides to solve for (1+r)(1 + r):(1+r)=(1.35)15(1 + r) = (1.35)^{\frac{1}{5}}We can use a calculator to find the fifth root of 1.351.35.
  4. Find Growth Rate: Using a calculator, we find that:\newline(1+r)≈1.062(1 + r) \approx 1.062\newlineThis means that 1+r1 + r is approximately 1.0621.062.
  5. Convert to Percentage: To find the annual growth rate rr, we subtract 11 from both sides:\newliner≈1.062−1r \approx 1.062 - 1\newliner≈0.062r \approx 0.062
  6. Convert to Percentage: To find the annual growth rate rr, we subtract 11 from both sides:\newliner≈1.062−1r \approx 1.062 - 1\newliner≈0.062r \approx 0.062To express the growth rate as a percentage, we multiply rr by 100100:\newlineAnnual growth rate ≈0.062×100\approx 0.062 \times 100\newlineAnnual growth rate ≈6.2%\approx 6.2\%

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