Question

a colony of bacteria decreases in population by 10 percent each hour. the initial bacteria culture contained 250 spores. what exponential equation can be used to find the terms in the geometric sequence? (1 point) \\(\bigcirc\\) \\(b(x) = 250(0.99)^{x - 1}\\) \\(\bigcirc\\) \\(b(x) = 250(0.90)^{x - 1}\\) \\(\bigcirc\\) \\(b(x) = 250 - 10(x - 1)\\) \\(\bigcirc\\) \\(b(x) = 250(0.10)^{x - 1}\\)

Explanation:

Step1: Recall Geometric Sequence Formula

The general formula for a geometric sequence is \( a_n = a_1(r)^{n - 1} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number. Here, \( x \) represents the term number, so the formula becomes \( b(x)=a_1(r)^{x - 1} \).

Step2: Identify Initial Term (\(a_1\))

The initial bacteria culture has 250 spores, so \( a_1 = 250 \).

Step3: Calculate Common Ratio (\(r\))

The population decreases by 10% each hour. A 10% decrease means the remaining population is \( 100\% - 10\% = 90\% \) of the previous hour's population. Converting 90% to a decimal gives \( r = 0.90 \).

Step4: Substitute \(a_1\) and \(r\) into the Formula

Substituting \( a_1 = 250 \) and \( r = 0.90 \) into \( b(x)=a_1(r)^{x - 1} \), we get \( b(x)=250(0.90)^{x - 1} \).

Answer:

\( \boldsymbol{b(x) = 250(0.90)^{x - 1}} \) (the second option)