Question

a growth medium is inoculated with 1,000 bacteria, which grow at a rate of 15% each day. what is the population of the culture 6 days after inoculation?

$y = 1,000(1.15)^{6}$; 2,313 bacteria
$y = 1,000(1.15)^{7}$; 2,660 bacteria
$y = 1,000(1.5)^{5}$; 7,594 bacteria
$y = 1,000(1.5)^{6}$; 11,391 bacteria

Explanation:

Step1: Identify the exponential growth formula

The formula for exponential growth is \(y = a(1 + r)^t\), where \(a\) is the initial amount, \(r\) is the growth rate, and \(t\) is the time. Here, \(a = 1000\), \(r=0.15\), and \(t = 6\).

Step2: Substitute the values into the formula

Substitute \(a = 1000\), \(r = 0.15\) (so \(1 + r=1.15\)), and \(t = 6\) into \(y=a(1 + r)^t\). We get \(y = 1000\times(1.15)^6\).

Step3: Calculate \((1.15)^6\)

\((1.15)^6=1.15\times1.15\times1.15\times1.15\times1.15\times1.15\approx2.313\).

Step4: Calculate \(y\)

\(y = 1000\times2.313 = 2313\).

Answer:

\(y = 1,000(1.15)^{6}; 2,313\) bacteria (the first option).