when t=0, Q=0.414 = a
e^.885 = 2.423, so Q = .414(2.423^t)
population grows by a factor of 2.423 every period.
well, duh: k = 0.885 = 88.5%, though it's not usually expressed as a percent.
Actually, that's k if you express Q as a e^kt
Round the answer for r to two decimal places
Q = 0.414(e^0.885)^t
a = ?
r = ?%
k = ?%
I found a to be 0.414 but when finding the percentage for r and k I got stuck.
e^.885 = 2.423, so Q = .414(2.423^t)
population grows by a factor of 2.423 every period.
well, duh: k = 0.885 = 88.5%, though it's not usually expressed as a percent.
Actually, that's k if you express Q as a e^kt
For the equation Q = 0.414(e^0.885)^t, we need to find the values of a, r, and k.
First, you correctly found that the starting value, a, is 0.414. Great job!
Now, let's find the growth rate, r. In this case, we have (0.885)^t. To find the growth rate, we need to calculate the exponent first.
If you round 0.885 to two decimal places, we get 0.89. So the equation becomes (0.89)^t.
Now, we know that the continuous growth rate, k, is equal to ln(r), where ln denotes the natural logarithm. So, to find k, we need to find r first.
If we rearrange the equation (0.89)^t = r, we can take the logarithm of both sides. Let's use the natural logarithm:
ln((0.89)^t) = ln(r)
Applying the logarithmic properties, we get:
t * ln(0.89) = ln(r)
Finally, divide both sides by t:
ln(r) = t * ln(0.89)
Now, we can substitute the values in to find r and k.
Let's say t = 1 (just as an example). Plug it into the equation:
ln(r) = 1 * ln(0.89)
Now, use a calculator to find ln(0.89) and multiply it by 1. This will give you the value of ln(r).
To convert ln(r) back to r, you'll need to take the inverse natural logarithm, e^(ln(r)), using a calculator. This will give you r.
Once you have r, you can find k by taking ln(r).
Remember to round r to two decimal places, as specified.
I hope this helps! Let me know if you need any more guidance or if you'd like to hear a joke while waiting for your calculations.
For the growth rate r:
r = (e^(0.885) - 1) * 100
For the continuous growth rate k:
k = ln(e^(0.885))
Let's calculate these values step by step.
Step 1: Calculate r
r = (e^(0.885) - 1) * 100
r = (e^(0.885) - 1) * 100
r ≈ (2.422 - 1) * 100
r ≈ 1.422 * 100
r ≈ 142.2
So, the growth rate r is approximately 142.2%.
Step 2: Calculate k
k = ln(e^(0.885))
k = 0.885
So, the continuous growth rate k is 0.885.
Therefore, the final values are:
a = 0.414
r ≈ 142.2%
k = 0.885
Q = 0.414 * 1
Thus, the starting value (a) is equal to 0.414.
Now let's move on to finding the growth rate (r) as a percentage. In the given equation, we can see that the growth rate (r) is represented by the exponent of the base e.
We have r = 0.885. To convert this decimal into a percentage, we can multiply it by 100.
r = 0.885 * 100
Calculating this expression, we find:
r = 88.5%
Hence, the growth rate (r) is 88.5%.
Finally, let's determine the continuous growth rate (k) as a percentage. The continuous growth rate (k) is related to the growth rate (r) by the formula k = (e^r - 1) * 100.
Using the rounded value of r (88.5%), we can find k as follows:
k = (e^(0.885) - 1) * 100
Evaluating this expression, we get:
k ≈ 141.25%
Therefore, the continuous growth rate (k) is approximately 141.25%.