A scientist counts 25 bacteria present in a culture and finds that the number of bacteria triplets each hour. The function y=25•3^x models the number of bacteria after x hours. Estimate when there will be about 1170 bacteria in the culture.
To estimate when there will be about 1170 bacteria in the culture, we need to solve the equation 1170 = 25 • 3^x.
Divide both sides of the equation by 25 to isolate 3^x: 1170/25 = 3^x
Simplify the left side: 46.8 = 3^x
Now we need to find the value of x that satisfies this equation. Taking the logarithm of both sides of the equation will help us do this. We can use the natural logarithm (ln) or the common logarithm (log) for this calculation.
Using the natural logarithm (ln):
ln(46.8) = ln(3^x)
Using the logarithm power rule: x • ln(3) = ln(46.8)
Divide both sides of the equation by ln(3): x = ln(46.8) / ln(3)
Using a calculator, we find: x ≈ 4.016
Therefore, it is estimated that there will be about 1170 bacteria in the culture after approximately 4.016 hours.