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A company has the goal of doubling their revenue from xx million dollars to 2x2x million dollars over the next nn years. If the company increases its revenue by y%y\% in the first year, by approximately what percentage must the company increase their revenue in the second year in order to reach their goal?

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Q. A company has the goal of doubling their revenue from xx million dollars to 2x2x million dollars over the next nn years. If the company increases its revenue by y%y\% in the first year, by approximately what percentage must the company increase their revenue in the second year in order to reach their goal?
  1. Identify Initial and Goal Revenue: Identify the initial revenue and the goal revenue.\newlineLet's denote the initial revenue as RR million dollars. The goal is to double this to 2R2R million dollars over the next two years.
  2. Determine First Year Revenue: Determine the revenue after the first year.\newlineIf the company increases its revenue by a certain percentage in the first year, let's denote this percentage as P1P_1. The revenue after the first year will be R×(1+P1100)R \times (1 + \frac{P_1}{100}).
  3. Calculate Required Second Year Revenue: Calculate the required revenue after the second year to meet the goal.\newlineThe required revenue after the second year is the goal revenue, which is 2R2R million dollars.
  4. Set Up Equation for Percentage Increase: Set up the equation to find the percentage increase needed in the second year. Let's denote the percentage increase needed in the second year as P2P_2. The equation will be: R×(1+P1100)×(1+P2100)=2RR \times (1 + \frac{P_1}{100}) \times (1 + \frac{P_2}{100}) = 2R
  5. Solve for P22: Solve for P22.\newlineDivide both sides of the equation by RR to simplify:\newline(1+P1100)⋅(1+P2100)=2(1 + \frac{P1}{100}) \cdot (1 + \frac{P2}{100}) = 2\newlineNow, we need to isolate P2P2. First, let's calculate the factor by which the revenue needs to increase in the second year, which is 2(1+P1100).\frac{2}{(1 + \frac{P1}{100})}.
  6. Calculate Exact Value for P22: Calculate the exact value for P22.\newlineP2/100=(2/(1+P1/100))−1P2/100 = (2 / (1 + P1/100)) - 1\newlineP2=100×[(2/(1+P1/100))−1]P2 = 100 \times [(2 / (1 + P1/100)) - 1]\newlineWithout the exact value of P1P1, we cannot calculate the numerical value for P2P2. This is where the solution must end, as we do not have the specific percentage increase from the first year.

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