If there are initially 12,000 bacteria in a culture, and the number of bacteria doubles each hour, the number of bacteria after [tex]\(t\)[/tex] hours can be found using the formula [tex]\(N=12,000\left(2^t\right)\)[/tex]. How many bacteria will be present after 7 hours?

A. 384,000
B. 168,000
C. 768,000
D. 1,536,000



Answer :

To solve the problem of determining the number of bacteria in a culture after 7 hours, given that the initial number of bacteria doubles every hour, we use the exponential growth formula:

[tex]\[ N = 12{,}000 \left(2^t\right) \][/tex]

where:
- [tex]\(N\)[/tex] is the number of bacteria after [tex]\(t\)[/tex] hours,
- [tex]\(t\)[/tex] is the time in hours,
- the initial number of bacteria is [tex]\(12{,}000\)[/tex].

For this problem, [tex]\( t = 7 \)[/tex] hours. We need to substitute [tex]\( t = 7 \)[/tex] into the formula and simplify.

Substituting [tex]\( t = 7 \)[/tex] into the formula, we get:
[tex]\[ N = 12{,}000 \left(2^7\right) \][/tex]

First, we calculate [tex]\( 2^7 \)[/tex]:
[tex]\[ 2^7 = 128 \][/tex]

Now, substitute [tex]\( 128 \)[/tex] back into the equation:
[tex]\[ N = 12{,}000 \times 128 \][/tex]

Multiplying these together:
[tex]\[ N = 1{,}536{,}000 \][/tex]

Therefore, the number of bacteria present after 7 hours is [tex]\( 1{,}536{,}000 \)[/tex].

The correct answer is [tex]\( 1{,}536{,}000 \)[/tex].