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: Recall 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 (as a decimal), and \( t \) is the time. Here, \( a = 1000 \), \( r = 0.15 \) (since 15% = 0.15), so \( 1 + r = 1.15 \), and \( t = 6 \) (days after inoculation). So the formula should be \( y = 1000(1.15)^6 \).

Step2: Calculate \( (1.15)^6 \)

First, calculate \( 1.15^6 \). Let's compute step by step:
\( 1.15^2 = 1.3225 \)
\( 1.15^3 = 1.15\times1.3225 = 1.520875 \)
\( 1.15^4 = 1.15\times1.520875 = 1.74900625 \)
\( 1.15^5 = 1.15\times1.74900625 = 2.0113571875 \)
\( 1.15^6 = 1.15\times2.0113571875 \approx 2.313060765625 \)

Step3: Multiply by the initial amount

Multiply by \( a = 1000 \): \( y = 1000\times2.313060765625 \approx 2313 \)

Answer:

A. \( y = 1000(1.15)^6 \); 2,313 bacteria