E Exponential and Logarithmic Functions Writing an equation that models exponential growth or decay To begin a bacteria study, a petri dish had 2000 bacteria cells. Each hour since, the number of cells has increased by \( 17 \% \). Sydney Write an exponential function showing the relationship between \( y \) and \( t \).

To model the exponential growth of bacteria in the petri dish, we can use the general formula for exponential growth: \[ y = y_0 \cdot (1 + r)^t \] where: - \( y \) is the amount of bacteria at time \( t \), - \( y_0 \) is the initial amount of bacteria, - \( r \) is the growth rate (as a decimal), - \( t \) is the time in hours. ### Step 1: Identify the known values - The initial number of bacteria, \( y_0 = 2000 \). - The growth rate, \( r = 17\% = 0.17 \). ### Step 2: Substitute the known values into the formula Now we can substitute \( y_0 \) and \( r \) into the formula: \[ y = 2000 \cdot (1 + 0.17)^t \] ### Step 3: Simplify the equation Calculating \( 1 + 0.17 \): \[ 1 + 0.17 = 1.17 \] So the equation becomes: \[ y = 2000 \cdot (1.17)^t \] ### Final Exponential Function The exponential function that models the growth of the bacteria is: \[ y = 2000 \cdot (1.17)^t \] This equation shows the relationship between the number of bacteria \( y \) and the time \( t \) in hours.