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P(t)=25(2)^((t)/( 1.06)) The number of yeast cells, P(t), in a culture after t days is modeled by the equation shown. After how many days will the population double in size? (Round your answer to the nearest hundredth.)

P(t)=25(2)t1.06 P(t)=25(2)^{\frac{t}{1.06}} \newlineThe number of yeast cells, P(t) P(t) , in a culture after t t days is modeled by the equation shown. After how many days will the population double in size?\newline(Round your answer to the nearest hundredth.)

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Q. P(t)=25(2)t1.06 P(t)=25(2)^{\frac{t}{1.06}} \newlineThe number of yeast cells, P(t) P(t) , in a culture after t t days is modeled by the equation shown. After how many days will the population double in size?\newline(Round your answer to the nearest hundredth.)
  1. Identify initial population and doubling condition: Identify the initial population and the condition for doubling.\newlineThe initial population is given by P(0)P(0), which is 2525 since t=0t=0 makes the exponent zero, and 20=12^0 = 1. To double, the population needs to reach 2×25=502 \times 25 = 50.
  2. Set up equation for population doubling: Set up the equation to solve for the time t when the population doubles.\newlineWe need to find t such that P(t) = 5050. So we set up the equation 5050 = 2525(22)^{\frac{t}{11.0606}}.
  3. Isolate exponential part of the equation: Divide both sides of the equation by 2525 to isolate the exponential part.\newline5025=25(2)(t1.06)25\frac{50}{25} = \frac{25(2)^{\left(\frac{t}{1.06}\right)}}{25} simplifies to 2=2(t1.06)2 = 2^{\left(\frac{t}{1.06}\right)}.
  4. Take logarithm of both sides: Take the logarithm of both sides to solve for tt.\newlineUsing the property of logarithms that log⁡b(ax)=xlog⁡b(a)\log_b(a^x) = x \log_b(a), we apply the logarithm base 22 to both sides: log⁡2(2)=log⁡2(2(t1.06))\log_2(2) = \log_2(2^{(\frac{t}{1.06})}).
  5. Simplify logarithmic equation: Simplify the logarithmic equation.\newlineSince log⁡2(2)=1\log_2(2) = 1, we have 1=t1.061 = \frac{t}{1.06}. Now we just need to solve for tt.
  6. Multiply both sides to find tt: Multiply both sides by 1.061.06 to find the value of tt.1×1.06=t1.06×1.061 \times 1.06 = \frac{t}{1.06} \times 1.06 gives us t=1.06t = 1.06.
  7. Round answer to nearest hundredth: Round the answer to the nearest hundredth as instructed.\newlineThe value of tt is already at the hundredth place, so rounding is not necessary. t=1.06t = 1.06.

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