To estimate the time, \( x \), when there will be approximately 550 bacteria in the culture, we can use the given function \( y = 35 \cdot 3^x \). We need to solve the equation:
\[ 550 = 35 \cdot 3^x \]
First, divide both sides by 35 to isolate \( 3^x \):
\[ \frac{550}{35} = 3^x \]
\[ \frac{550}{35} \approx 15.71 \]
So, we have:
\[ 15.71 \approx 3^x \]
To solve for \( x \), we take the logarithm of both sides. Using the natural logarithm (ln) for simplicity:
\[ \ln(15.71) = \ln(3^x) \]
Using the property of logarithms, \( \ln(a^b) = b\ln(a) \):
\[ \ln(15.71) = x \cdot \ln(3) \]
Now, solve for \( x \):
\[ x = \frac{\ln(15.71)}{\ln(3)} \]
Using the approximate values \( \ln(15.71) \approx 2.754 \) and \( \ln(3) \approx 1.098 \):
\[ x \approx \frac{2.754}{1.098} \approx 2.51 \]
So, \( x \approx 2.5 \), meaning that there will be about 550 bacteria in the culture after approximately 2.5 hours. Therefore, the correct answer is:
**about 2.5 hours.**
5.
A scientist counts 35 bacteria present in a culture and finds that the number of bacteria triples each hour. The function y = 35 ∙ 3x models the number of bacteria after x hours. Estimate when there will be about 550 bacteria in the culture.
about 5.5 hours
about 3.5 hours
about 2.5 hours
about 4.5 hours
1 answer