Write and solve the differential equation that models the verbal statement.
The rate of change of N is proportional to N. (Use k for the proportionality constant.)
dN
dt
=
N(t) =
I mainly need help with the bottom part below this sentence.
Evaluate the solution at the specified value of the independent variable.
When t = 0, N = 250, and when t = 1, N = 300. What is the value of N when t = 4?
N(4) =
so, you have
dN/dt = kN
dN/N = k dt
log N = kt + log c
N = c*e^(kt)
Now plug in the points (0,250) and (1,300) to find c and k
250 = c*1, so c=250
N = 250e^(kt)
similarly to find k.
Then use the result to find N(4)
To me that don’t make sense
Alright, let's start by writing the differential equation:
dN/dt = kN
Now, let's solve this differential equation:
Separate the variables by dividing both sides of the equation by N:
dN/N = k dt
Integrate both sides:
∫ dN/N = ∫ k dt
ln|N| = kt + C
where C is the constant of integration.
Exponentiate both sides:
e^(ln|N|) = e^(kt + C)
|N| = e^(kt) * e^C
Simplify using the absolute value:
|N| = Ce^(kt)
Now, we can use the given initial conditions to find the specific value of C. When t = 0, N = 250:
|250| = C * e^(k * 0)
250 = C
Therefore, our particular solution is:
N = 250e^(kt)
Now, let's evaluate the solution when t = 4:
N(4) = 250e^(k * 4)
Unfortunately, without further information or another initial condition, we cannot find the exact value of N at t = 4.
To derive the differential equation using the given information, we can express the verbal statement mathematically:
The rate of change of N is proportional to N.
This can be written as:
dN/dt = kN
Now, let's solve this differential equation using separation of variables:
Separate the variables by dividing both sides by N:
dN/N = k dt
Integrate both sides with respect to their respective variables:
∫(1/N) dN = ∫k dt
ln(N) = kt + C
where C is the constant of integration.
Now, let's solve for C using the initial condition N(0) = 250:
ln(250) = k * 0 + C
C = ln(250)
So the equation becomes:
ln(N) = kt + ln(250)
N = e^(kt + ln(250))
Now, we can evaluate the solution at t = 4 using the given initial condition N(1) = 300:
N(4) = e^(k*4 + ln(250))
To find the value of k, we can use N(1) = 300:
300 = e^(k*1 + ln(250))
Taking the natural logarithm of both sides:
ln(300) = k*1 + ln(250)
Solving for k:
k = ln(300) - ln(250)
Substituting the value of k back into the equation:
N(4) = e^((ln(300) - ln(250))*4 + ln(250))
Now, we can calculate the value of N(4) using the given values.
To solve the differential equation, we start by expressing the verbal statement mathematically:
The rate of change of N is proportional to N.
This can be written as:
dN/dt = kN
where dN/dt represents the rate of change of N with respect to time.
To solve this equation, we can separate the variables and then integrate both sides:
(dN/N) = k dt
Integrating both sides gives:
ln|N| = kt + C
where C is the constant of integration.
Next, we need to find the specific solution to the differential equation by using the given initial condition. We know that when t = 0, N = 250. Plugging these values into the equation, we get:
ln|250| = k(0) + C
ln|250| = C
Therefore, our specific solution becomes:
ln|N| = kt + ln|250|
Now, we need to evaluate the solution at t = 4. Plugging this value into our specific solution, we get:
ln|N| = k(4) + ln|250|
We don't have the value of k yet, so we need to find it using the second given condition when t = 1, N = 300. Plugging these values into the specific solution, we get:
ln|300| = k(1) + ln|250|
Now, we can solve this equation for k. Subtracting ln|250| from both sides gives:
ln|300| - ln|250| = k
Using a logarithm property, we can simplify it further:
ln|300/250| = k
ln|6/5| = k
Now that we have the value of k, we can substitute it back into our specific solution:
ln|N| = (ln|6/5|)t + ln|250|
Finally, we can evaluate N(4) by plugging t = 4 into the specific solution:
ln|N(4)| = (ln|6/5|)(4) + ln|250|
To find N(4), we need to exponentiate both sides:
N(4) = e^[(ln|6/5|)(4) + ln|250|]
Using the properties of logarithms, this can be further simplified:
N(4) = (e^ln|6/5|)^4 * e^ln|250|
N(4) = (6/5)^4 * 250
By evaluating this expression, we can find the value of N when t = 4.