A yeast grows at a rate proportional to its present size.if the original amount doubles in two hours in how many hours will it triple
A yeast grows at a rate proportional to its present size
----> amount = a e^(kt), where a is the initial amount and t is in hours
given: when t = 2, a = 1, and Amount = 2
2 = 1 e^(2k)
take ln of both sides and use log rules
ln2 = 2k
k = ln2/2
amount = 1 e^((ln2/2)t)
for tripling:
3 = e^ ((ln2/2)t) , solve for t, let me know what you get
Ah, yeast, nature's little multiplication machines! Let's calculate this with a pinch of humor, shall we?
If the original amount doubles in two hours, it means the yeast is growing exponentially. So, if it takes two hours to double, we can expect it to continue multiplying like rabbits on turbo mode.
Now, tripling the original amount means that it will need to go through an extra round of comedic yeast reproduction. Since it doubles every two hours, we can expect the yeast to triple in... *drumroll*... three hours!
So, in three hours, you'll have one heck of a yeast party on your hands. Just make sure they don't start telling yeast jokes because those tend to fall a bit flat!
To solve this problem, we can use the concept of exponential growth.
Let's say the original amount of yeast is denoted by A0. According to the problem, this original amount doubles in size in two hours. Therefore, after two hours, the size of the yeast will be 2 * A0.
Now, let's think about how long it takes for the yeast to triple in size. We know that the growth is proportional to its present size, so we can set up the following equation:
2 * A0 * e^(k * 2) = 3 * A0,
where e is the base of the natural logarithm (approximately 2.71828), and k is the constant of proportionality.
Simplifying the equation, we have:
e^(2k) = 3/2.
Now, take the natural logarithm of both sides of the equation:
ln(e^(2k)) = ln(3/2).
2k = ln(3/2).
Finally, divide both sides of the equation by 2 to solve for k:
k = (1/2) * ln(3/2).
Now that we have the value of k, we can determine how long it takes for the yeast to triple in size. Let's denote this time as t.
Using the growth equation, we can write:
A = A0 * e^(k * t).
Plugging in the value of A = 3 * A0 (since we want to triple the original amount), and rearranging the equation, we get:
3 = e^(k * t).
Taking the natural logarithm of both sides, we have:
ln(3) = k * t.
Solving for t, we get:
t = ln(3) / k.
Substituting the value of k we found earlier, we get:
t = ln(3) / [(1/2) * ln(3/2)].
Simplifying the expression, we have:
t = 2 * ln(3).
Therefore, it will take approximately 2 * ln(3) hours for the yeast to triple in size.
To determine the time it takes for the yeast to triple in size, we can use the concept of exponential growth. Since the yeast grows at a rate proportional to its present size, we can express this growth mathematically as:
y(t) = y₀ * e^(kt)
Where:
- y(t) represents the size of the yeast at time t.
- y₀ represents the initial size of the yeast.
- e is the mathematical constant approximately equal to 2.71828.
- k represents the growth rate.
- t represents time in hours.
In this case, we know that the yeast's original amount doubles in two hours. This means that after two hours, the size of the yeast will be twice its original size, represented as:
2y₀ = y₀ * e^(k * 2)
To determine k, we can rearrange the equation:
2 = e^(k * 2)
Taking the natural logarithm (ln) of both sides, we get:
ln(2) = k * 2
Since we only want to know the time it takes for the yeast to triple, we can solve for t when the size is three times the original:
3y₀ = y₀ * e^(kt)
3 = e^(kt)
Taking the natural logarithm (ln) of both sides:
ln(3) = kt
Now, let's solve for t:
t = ln(3) / k
We already calculated k as ln(2)/2, so substituting this value into the equation:
t = ln(3) / (ln(2) / 2)
Simplifying the expression:
t = 2 * ln(3) / ln(2)
Approximately, this value is:
t ≈ 2.585 hours
Therefore, it will take approximately 2.585 hours for the yeast to triple in size.