During a research experiment, it was found that the number of bacteria in a culture grew at a rate proportional to its size. At 7:00 AM there were 6,000 bacteria present in the culture. At noon, the number of bacteria grew to 6,500. How many bacteria will there be at midnight?

Answer: 7877

-------------------------------------
-------------------------------------

Work Shown:

x = elapsed time (in hours) since 7:00 AM
y = bacteria population in thousands
eg: something like y = 2 means 2000 bacteria

At 7:00 AM, x = 0 and y = 6
y = a*b^x
6 = a*b^0
6 = a*1
6 = a
a = 6

At 12:00 PM noon, which is 5 hours after 7:00 AM, the population is 6500.
So (x,y) = (5,6.5)
y = a*b^x
y = 6*b^x
6.5 = 6*b^5
6.5/6 = b^5
1.083333 = b^5
b^5 = 1.083333
b = (1.083333)^(1/5)
b = 1.016137
which is approximate

The approximate exponential growth model is 
y = 6*1.016137^x

Plug in x = 17 to find the population at midnight. Keep in mind that midnight is 12 hours after 12:00 PM noon, so 12 additional hours elapse after the initial 5, meaning a total of 12+5 = 17 hours pass by from 7:00 AM to 12:00 AM the next day. 

y = 6*1.016137^x
y = 6*1.016137^17
y = 7.876618

I made y be the population in thousands, so you'll need to multiply that y value by 1000 to get
7.876618*1000 = 7876.618 which rounds to 7877 when rounding to the nearest whole number

So that's why the answer is roughly 7877