I have this hard question in math.
can someone help me find out the riddle?
I start with 1 bacteria. Every hour it doubles.
How many hours until there are 1,000,000 bacteria?
Can you make a function that describes this situation?
What if I double it? I start with 2 glasses with 1 bacteria each?
NOW how many hours until there are 1,000,000 bacteria?
I am starting with twice as many, so will it be half the time?
(I am stumped)
actually, you start with one bacterium.
You need a function that doubles every time you add 1 to the time. You probably already are aware that this models exponential growth, so what such function doubles as needed? 2^x does the job.
So, starting at time x=0, at time x hours later, there will be 2^x bacteria.
So, when does 2^x = 1000000?
when x = log1000000/log2 = 6/log2
For the next part, your reasoning is correct.
Thankyou so much:)
To solve the problem, we can create a function to describe the situation. Let's call the number of bacteria at any given hour "n" and the number of hours it takes to reach 1,000,000 bacteria "h".
In the first scenario, we start with 1 bacteria and it doubles every hour. So, each hour the number of bacteria doubles. We can express this as:
n = 1 * 2^h
To find the number of hours it takes to reach 1,000,000 bacteria, we can solve the equation:
1,000,000 = 1 * 2^h
To solve for h, take the logarithm base 2 of both sides:
log2(1,000,000) = log2(2^h)
log2(1,000,000) = h
Using a calculator, we find that log2(1,000,000) is approximately equal to 19.93. Therefore, it takes approximately 20 hours for the number of bacteria to reach 1,000,000.
In the second scenario, we start with 2 glasses, each containing 1 bacteria. So, initially, we have 2 bacteria. The function becomes:
n = 2 * 2^h
Similarly, we can set up the equation:
1,000,000 = 2 * 2^h
Again, take the logarithm base 2 of both sides:
log2(1,000,000/2) = log2(2^h)
log2(500,000) = h
Using a calculator, we find that log2(500,000) is approximately equal to 18.93. Therefore, it takes approximately 19 hours for the number of bacteria to reach 1,000,000.
So, starting with twice as many bacteria does not result in half the time. The difference in time is only about 1 hour.
To find the number of hours it takes for the bacteria to reach 1,000,000, we can create a function that describes the situation. Let's call the number of bacteria at any given time 'b' and the number of hours as 'h'.
In the first scenario where you start with 1 bacteria and it doubles every hour, the function can be written as:
b = 1 * 2^h
This means that after 'h' hours, the number of bacteria will be 1 times 2 raised to the power of 'h'.
To find out how many hours it takes for the number of bacteria to reach 1,000,000, we can substitute 'b' with 1,000,000 in the equation and solve for 'h':
1,000,000 = 1 * 2^h
To simplify the equation, we can rewrite 1,000,000 as 10^6:
10^6 = 2^h
We can now take the logarithm of both sides with base 2:
log2(10^6) = log2(2^h)
6 = h * log2(2)
Since log2(2) is equal to 1, we get:
h = 6
So, it takes 6 hours for the number of bacteria to reach 1,000,000 in the first scenario.
In the second scenario where you start with 2 glasses with 1 bacteria each, you are essentially doubling the initial number. Since doubling the initial number doubles the growth rate, it will not be half the time. In this case, we can modify the function to:
b = 2 * 2^h
Repeat the above steps by substituting 'b' with 1,000,000:
1,000,000 = 2 * 2^h
We can rewrite this equation as:
2^21 = 2^h
The bases are equal, so the exponents must be equal:
21 = h
Therefore, it takes 21 hours for the number of bacteria to reach 1,000,000 in the second scenario. As you can see, it is not half the time because the growth rate still doubles every hour, regardless of the initial number of bacteria.
I hope this explanation helps! Let me know if you have any further questions.