\[
P(t)=25(2)^{\frac{t}{1.06}}
\]
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.)
Final Answer: \(\boxed{1.06}\)
Step 1 :The number of yeast cells, \(P(t)\), in a culture after \(t\) days is modeled by the equation \(P(t)=25(2)^{\frac{t}{1.06}}\). We want to find out after how many days will the population double in size.
Step 2 :The population doubles when \(P(t) = 2P(0)\). We know that \(P(0) = 25\), so we need to solve the equation \(25(2)^{\frac{t}{1.06}} = 2*25\) for \(t\).
Step 3 :We can simplify this equation to \(2^{\frac{t}{1.06}} = 2\), which implies that \(\frac{t}{1.06} = 1\). Solving for \(t\) will give us the number of days it takes for the population to double.
Step 4 :\(t = 1.06\)
Step 5 :Final Answer: \(\boxed{1.06}\)