Question

fiona started her study of bacteria growth with 1,000 bacteria in a petri dish. after 1 hour, the count was increased to 1,800. after 2 hours, the count was 3,240. after the third hour showed a count of 5,832, fiona wrote an equation to represent the exponential growth of the bacteria. what exponential growth function did fiona write?
$f(x) = \boldsymbol{▼} (\boldsymbol{▼})^{x}$

Explanation:

Step1: Recall Exponential Growth Formula

The general form of an exponential growth function is \( f(x) = a \cdot b^x \), where \( a \) is the initial amount, \( b \) is the growth factor, and \( x \) is the time (in hours here).

Step2: Identify Initial Amount (\(a\))

Fiona started with 1,000 bacteria, so \( a = 1000 \).

Step3: Calculate Growth Factor (\(b\))

After 1 hour (\(x = 1\)), the number of bacteria is 1,800. Substitute into the formula: \( 1800 = 1000 \cdot b^1 \). Solve for \( b \): \( b=\frac{1800}{1000}=1.8 \). We can verify with other times: After 2 hours, \( 1000 \cdot (1.8)^2 = 1000 \cdot 3.24 = 3240 \), which matches. After 3 hours, \( 1000 \cdot (1.8)^3 = 1000 \cdot 5.832 = 5832 \), which also matches.

Answer:

\( f(x) = 1000 \cdot (1.8)^x \)