At your initial meeting with your business partners for CellGenTech, you propose that the rate of change in the number of newly introduced cell lines that the company scientists should be able to produce should be proportional to the number of cell lines. The company initially patented and owned three cell lines N(0)=3 and six months later were able to pick up another two lines N(6)=5. After one year, the model goes awry. It poorly predicts the number of lines. So, the initial model is changed to reveal that the rate of change in the number of cell lines are proportional to the square of the number of cell lines.
If the old model remained, what percentage error would you find after one year in the predicted and actual value? Note: Round in the tenths place and enter the percent value but do not include the percent symbol (eg. for 10.483% enter 10.5). Also, to find the percent error, use
\%e = 100*|A-E|/A where A is the actual value and E is the experimental value (or first model value).
I wasn't sure where to start on this one.
To solve this problem, let's start with the given information. The old model states that the rate of change in the number of cell lines, denoted as dN/dt, is proportional to the number of cell lines, N.
Therefore, we can write the differential equation as:
dN/dt = kN
where k is the proportionality constant.
We can solve this first-order ordinary differential equation by separation of variables:
1/N dN = k dt
Integrating both sides:
∫ (1/N) dN = ∫ k dt
ln(N) = kt + C
Where C is the constant of integration.
Now, we can solve for C using the initial condition N(0) = 3:
ln(3) = k * 0 + C
C = ln(3)
Substituting C back into the equation:
ln(N) = kt + ln(3)
Exponentiating both sides:
N = e^(kt + ln(3))
Next, we can use the information that after six months (0.5 years), the number of cell lines is 5 (N(0.5) = 5):
N(0.5) = e^(k * 0.5 + ln(3))
5 = e^(0.5k + ln(3))
Now, we need to find the value of k by solving this equation.
Divide both sides by e^(ln(3)):
5 / 3 = e^(0.5k)
Taking the natural logarithm of both sides:
ln(5/3) = 0.5k
Solving for k:
k = 2 * ln(5/3)
Now that we have the value of k, we can calculate the predicted value for N after one year (N(1)) using the old model:
N(1) = e^(k * 1 + ln(3))
N(1) = e^(2 * ln(5/3) + ln(3))
N(1) ≈ 3.828
The actual value of the number of cell lines after one year is given as N_actual = 9.
Now we can calculate the percentage error using the formula:
% error = 100 * |N_actual - N_predicted| / N_actual
% error = 100 * |9 - 3.828| / 9
% error ≈ 57.5
Therefore, the percentage error, if the old model remained, would be approximately 57.5%.
To find the percentage error between the predicted value and the actual value using the old model, you need to first determine the predicted value of the number of cell lines after one year.
In the initial model, the rate of change in the number of cell lines is proportional to the number of cell lines. Mathematically, this can be expressed as:
dN/dt = kN
where dN/dt represents the rate of change of the number of cell lines over time, N represents the number of cell lines, and k represents the proportionality constant.
To solve this first-order differential equation, you can separate variables and integrate:
1/N dN = k dt
∫ 1/N dN = ∫ k dt
ln|N| = kt + C
N = e^(kt+C)
N = Ce^kt
Using the given initial condition N(0) = 3, you can substitute the values and solve for C:
3 = Ce^0
C = 3
So, the equation for the number of cell lines predicted by the old model is:
N(t) = 3e^kt
Next, you need to determine the k value. You are given that after six months (t = 6), the number of cell lines is 5 (N(6) = 5). You can substitute these values and solve for k:
5 = 3e^(6k)
e^(6k) = 5/3
6k = ln(5/3)
k = (1/6)ln(5/3)
Now you have the equation for the number of cell lines predicted by the old model, N(t) = 3e^((1/6)ln(5/3)t).
To find the predicted number of cell lines after one year (t = 12), you can substitute t = 12 into the equation:
N(12) = 3e^((1/6)ln(5/3) * 12)
Using a calculator, you can evaluate this expression to find the predicted number of cell lines after one year.
Once you have the predicted value, you can calculate the percentage error using the formula:
%e = 100 * |A - E| / A
where A is the actual value (given) and E is the experimental value (predicted by the old model).
Let's go ahead and calculate the predicted value and the percentage error together.