Question

1. a bacteria culture starts with 150 cells and grows by 25% every hour.

a) write the exponential equation to represent the number of bacteria cells after t hours.
b) find the population after 3 hours.

Explanation:

Step1: Determine the general form of the exponential growth equation

The general form of an exponential growth equation is \(y = a(1 + r)^t\), where \(a\) is the initial amount, \(r\) is the growth rate as a decimal, and \(t\) is the time.
Given \(a = 150\) (initial number of cells) and \(r=0.25\) (since \(25\%=0.25\)), the equation is \(y = 150(1 + 0.25)^t\).

Step2: Simplify the equation

Simplify \(1+0.25 = 1.25\), so the equation for part (a) is \(y = 150\times1.25^t\).

Step3: Calculate the population after \(t = 3\) hours for part (b)

Substitute \(t = 3\) into the equation \(y = 150\times1.25^t\).
First, calculate \(1.25^3=1.25\times1.25\times1.25=\frac{5}{4}\times\frac{5}{4}\times\frac{5}{4}=\frac{125}{64}\approx1.953125\).
Then, \(y = 150\times1.953125 = 292.96875\approx293\) (since we can't have a fraction of a cell in this context).

Answer:

a) \(y = 150\times1.25^t\)
b) \(293\)