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12 Jan 2026, 10:12
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A culture of bacteria had a population of 8,000 at 10 AM. The population grew at a fixed percentage rate per hour and reached 64,000 at 7 PM on the same day. If the population continues to grow at that same rate, at what time will the population reach 256,000?
A. 9 PM of the same day
B. 10 PM of the same day
C. 12 AM of the next day
D. 12:30 AM of the next day
E. 1 AM of the next day
A. 9 PM of the same day
B. 10 PM of the same day
C. 12 AM of the next day
D. 12:30 AM of the next day
E. 1 AM of the next day
12 Jan 2026, 10:12
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Official Solution:
A culture of bacteria had a population of 8,000 at 10 AM. The population grew at a fixed percentage rate per hour and reached 64,000 at 7 PM on the same day. If the population continues to grow at that same rate, at what time will the population reach 256,000?
A. 9 PM of the same day
B. 10 PM of the same day
C. 12 AM of the next day
D. 12:30 AM of the next day
E. 1 AM of the next day
In 9 hours, from 10 AM to 7 PM, the population grew by a factor of 8 (64,000/8,000 = 8).
Since the rate of growth is constant, the population doubles every 3 hours: (2 * 2 * 2 = 8) over (3 + 3 + 3 = 9) hours. The same rate of increase ( * 2) in the same time period (3 hours).
Therefore, to grow 4 times (from 64,000 to 256,000), the population will need an additional 6 hours (3 hours + 3 hours).
Hence, the population will reach 256,000 at 7 PM + 6 hours = 1 AM of the next day.
Answer: E.
Alternatively, here is the algebraic approach.
Given:
\(8,000 * r^9 = 64,000\)
\(r = \sqrt[9]{8} = \sqrt[9]{2^3} = \sqrt[3]{2}\)
We need to find such \(x\), where \(x\) is the number of hours needed from 10 AM for the population to reach 256,000, such that:
\(8,000 * r^x = 256,000\)
\(r^x = 32\)
\((\sqrt[3]{2})^x = 32\)
\(2^{\frac{x}{3}} = 2^5\)
\(\frac{x}{3} = 5\)
\(x = 15\)
Thus, after 15 hours from 10 AM (which is 1 AM of the next day), the population will reach 256,000.
Answer: E
A culture of bacteria had a population of 8,000 at 10 AM. The population grew at a fixed percentage rate per hour and reached 64,000 at 7 PM on the same day. If the population continues to grow at that same rate, at what time will the population reach 256,000?
A. 9 PM of the same day
B. 10 PM of the same day
C. 12 AM of the next day
D. 12:30 AM of the next day
E. 1 AM of the next day
In 9 hours, from 10 AM to 7 PM, the population grew by a factor of 8 (64,000/8,000 = 8).
Since the rate of growth is constant, the population doubles every 3 hours: (2 * 2 * 2 = 8) over (3 + 3 + 3 = 9) hours. The same rate of increase ( * 2) in the same time period (3 hours).
Therefore, to grow 4 times (from 64,000 to 256,000), the population will need an additional 6 hours (3 hours + 3 hours).
Hence, the population will reach 256,000 at 7 PM + 6 hours = 1 AM of the next day.
Answer: E.
Alternatively, here is the algebraic approach.
Given:
\(8,000 * r^9 = 64,000\)
\(r = \sqrt[9]{8} = \sqrt[9]{2^3} = \sqrt[3]{2}\)
We need to find such \(x\), where \(x\) is the number of hours needed from 10 AM for the population to reach 256,000, such that:
\(8,000 * r^x = 256,000\)
\(r^x = 32\)
\((\sqrt[3]{2})^x = 32\)
\(2^{\frac{x}{3}} = 2^5\)
\(\frac{x}{3} = 5\)
\(x = 15\)
Thus, after 15 hours from 10 AM (which is 1 AM of the next day), the population will reach 256,000.
Answer: E
01 Apr 2026, 11:14
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It takes 9 hours to reach 8x ( from 8000 to 64000).
Having said that it has fixed rate. It translates that every 3 hours population is getting 2x .
Now we have to find how much time it will take to reach from 64000 to 256000 (which is 4x from 64000)
Now every 3 hours population is getting 2x then every 6 hours (3+3) population will get 4x.
So time to reach 256000 will be 7 PM + 6 Hours i.e 1 a.m of the next day
Having said that it has fixed rate. It translates that every 3 hours population is getting 2x .
Now we have to find how much time it will take to reach from 64000 to 256000 (which is 4x from 64000)
Now every 3 hours population is getting 2x then every 6 hours (3+3) population will get 4x.
So time to reach 256000 will be 7 PM + 6 Hours i.e 1 a.m of the next day
