a colony of bacteria living on a petri dish under optimal conditions doubles in size every ten minutes. At noon on a certain day, the Petri dish completely covered with bacteria. At what times (to the nearest hour, minute and second) was the percentage of the plate covered by bacteria: 50%, 25%, 5% or 1%
amount = c (2^(t/10)), where t is in minutes and c is the initial amount
so when covered by 50% ....
.5 = 1 (2^(t/10)
take log of both sides
log .5 = log 2^(t/10)
log .5 = (t/10)log 2
t/10 = log .5/log2 = -1
t = -10
So it was covered to 50% 10 minutes ago, or at 11:50 am
when covered at 5%
.05 = 1 (2^(t/10))
log .05 = log 2^t/10
t/10 = log .05/log2 = -4.32192..
t = -43.2192 minutes ago
at appr 11:16 am
do the others in the same way
Well, it seems the bacteria on the petri dish are partying like there's no tomorrow! Let me do some quick calculations and clown around with these numbers.
We know that the colony doubles in size every 10 minutes, so at noon it is fully covered. Now let's find out when it reaches those interesting percentages.
To calculate the time it takes to reach each percentage, we'll count the number of times the bacteria double until it hits the desired coverage.
For 50%, we need the colony to double once to fully cover half the plate. Since it doubles every 10 minutes, it will take 10 minutes to reach 50% coverage.
For 25%, the colony will need to double twice to cover a quarter of the plate. It will take 20 minutes (2 times 10) to reach 25% coverage.
For 5%, we need the colony to double four times. It will take 40 minutes (4 times 10) to reach 5% coverage.
And finally, for 1%, the colony will need to double seven times. It will take 70 minutes (7 times 10) to reach 1% coverage.
So, to summarize:
- To reach 50% coverage: 10 minutes past noon.
- To reach 25% coverage: 20 minutes past noon.
- To reach 5% coverage: 40 minutes past noon.
- To reach 1% coverage: 70 minutes past noon.
Just be careful not to accidentally breathe on the plate and blow them all away before they reach these percentages!
To find the times at which the petri dish was covered by certain percentages of bacteria, we can use exponential growth calculations.
Let's start by calculating the number of doubling times required for the bacteria colony to reach each percentage.
1. 50% coverage:
To reach 50% coverage, the colony would have to double its size once. Since it doubles every 10 minutes, it takes 10 minutes to reach 50% coverage.
2. 25% coverage:
To reach 25% coverage, the colony would have to double its size twice. Since it doubles every 10 minutes, it takes 20 minutes to reach 25% coverage.
3. 5% coverage:
To reach 5% coverage, the colony would have to double its size four times. Since it doubles every 10 minutes, it takes 40 minutes to reach 5% coverage.
4. 1% coverage:
To reach 1% coverage, the colony would have to double its size seven times. Since it doubles every 10 minutes, it takes 70 minutes to reach 1% coverage.
Now, let's calculate the times at which each percentage of coverage would be reached.
Assuming noon as the starting point:
1. 50% coverage:
Noon + 10 minutes = 12:10:00 PM
2. 25% coverage:
Noon + 20 minutes = 12:20:00 PM
3. 5% coverage:
Noon + 40 minutes = 12:40:00 PM
4. 1% coverage:
Noon + 70 minutes = 1:10:00 PM
So, the times (to the nearest hour, minute, and second) at which the petri dish was approximately covered by each percentage of bacteria are as follows:
- 50% coverage: 12:10:00 PM
- 25% coverage: 12:20:00 PM
- 5% coverage: 12:40:00 PM
- 1% coverage: 1:10:00 PM
To determine the times at which the percentage of the petri dish covered by bacteria reaches specific values, we need to calculate the time intervals it takes for the colony to grow from 100% to each given percentage.
Since the colony doubles in size every ten minutes, we can calculate the number of times it doubles to reach a specific percentage. Let's go step by step:
1. Find the number of times the colony doubles to reach 50% of the plate:
To get from 100% to 50%, the colony needs to go through one doubling. Since each doubling takes ten minutes, the colony will reach 50% after ten minutes.
Therefore, the time is 12:00 PM + 10 minutes = 12:10 PM.
2. Find the number of times the colony doubles to reach 25%:
To get from 100% to 25%, the colony needs to go through two doublings. Each doubling takes ten minutes, so we need to calculate the time for two doublings.
First doubling: 12:00 PM + 10 minutes = 12:10 PM.
Second doubling: 12:10 PM + 10 minutes = 12:20 PM.
Therefore, the time is 12:20 PM.
3. Find the number of times the colony doubles to reach 5%:
To get from 100% to 5%, the colony needs to go through four doublings. Each doubling takes ten minutes, so we need to calculate the time for four doublings.
First doubling: 12:00 PM + 10 minutes = 12:10 PM.
Second doubling: 12:10 PM + 10 minutes = 12:20 PM.
Third doubling: 12:20 PM + 10 minutes = 12:30 PM.
Fourth doubling: 12:30 PM + 10 minutes = 12:40 PM.
Therefore, the time is 12:40 PM.
4. Find the number of times the colony doubles to reach 1%:
To get from 100% to 1%, the colony needs to go through six doublings. Each doubling takes ten minutes, so we need to calculate the time for six doublings.
First doubling: 12:00 PM + 10 minutes = 12:10 PM.
Second doubling: 12:10 PM + 10 minutes = 12:20 PM.
Third doubling: 12:20 PM + 10 minutes = 12:30 PM.
Fourth doubling: 12:30 PM + 10 minutes = 12:40 PM.
Fifth doubling: 12:40 PM + 10 minutes = 12:50 PM.
Sixth doubling: 12:50 PM + 10 minutes = 1:00 PM.
Therefore, the time is 1:00 PM.
To summarize, the times at which the percentage of the petri dish covered by bacteria is 50%, 25%, 5%, and 1% are as follows:
50%: 12:10 PM
25%: 12:20 PM
5%: 12:40 PM
1%: 1:00 PM