Zack's biology lab group is studying how quickly bacteria grow when exposed to sunlight. At the beginning of the experiment, Zack looked at the sample under a microscope and counted 5 bacteria. After the sample was exposed to sunlight for one hour, Zack looked again and counted 35 bacteria.

Question

Write an exponential equation in the form y=a(b)x that can model the number of bacteria, y, after x hours in sunlight.
Use whole numbers, decimals, or simplified fractions for the values of a and b.

Answer 1

To model the growth of bacteria using an exponential equation in the form \( y = a(b)^x \), we need to identify the parameters \( a \) and \( b \) using the information provided.

1. Understanding the Parameters:

   • \( a \) is the initial amount of bacteria. From the problem, we know that initially (at \( x = 0 \)), there were 5 bacteria. So, \( a = 5 \).
   • \( b \) is the growth factor per hour.

2. Using the Information Given:

   • After 1 hour (\( x = 1 \)), the number of bacteria is 35. Therefore, we can set up the equation: \[ y = 5(b)^x \] For \( x = 1 \): \[ 35 = 5(b)^1 \]

3. Solving for \( b \):

   • We can simplify the equation: \[ 35 = 5b \] To find \( b \), divide both sides by 5: \[ b = \frac{35}{5} = 7 \]

4. Writing the Final Equation:

   • Now we have \( a = 5 \) and \( b = 7 \). Thus, the exponential model for the number of bacteria \( y \) after \( x \) hours in sunlight is: \[ y = 5(7)^x \]

So, the exponential equation that models the bacteria growth is: \[ y = 5(7)^x \]