At a summer campfire, the radius of a marshmallow on a stick expands at the rate of
\ {r ' (t)} = \frac{2.1 }{1 t + 5} mm/s
where t is the time of heating in seconds. Initially the radius was 3.8 mm. Find the radius after 27 seconds using the following steps:
When substituting, du = dt
The resulting integral looks like \int \frac{A}{u} \, du where A is a numerical value.
A =
The constant of integration, C is
The radius after 27 seconds is
mm
r' = 2.1/(t+5)
r = 2.1 log(t+5) + C
at t=0,
3.8 = 2.1 log5 + C
C = 3.8-2.1log5 = 0.42
so,
r = 2.1 log(t+5)+0.42
now just plug in t=27
To find the radius after 27 seconds, we need to integrate the given derivative of the radius with respect to time.
Step 1: Set up the integration:
We are given the derivative \(\frac{dr}{dt} = \frac{2.1}{1t + 5} \, \text{mm/s}\).
To integrate this derivative, we can use the substitution method.
Let \(u = 1t + 5\), then \(du = dt\).
Step 2: Substitute the variables:
Substitute \(u\) for \(1t + 5\) and \(du\) for \(dt\) in the given derivative to get:
\(\frac{dr}{du} = \frac{2.1}{u} \, \text{mm/s}\).
Step 3: Integrate:
Integrating the above expression will give us the radius function:
\(\int dr = \int \frac{2.1}{u} du\).
Step 4: Apply the integral:
The integral of \(\frac{1}{u}\) with respect to \(u\) is given by \(\ln|u| + C\), where \(C\) is the constant of integration.
So, the integral becomes:
\(r = 2.1 \ln|u| + C\).
Step 5: Substitute back the value of \(u\):
Substitute \(u = 1t + 5\) back into the radius function:
\(r = 2.1 \ln|1t + 5| + C\).
Step 6: Find the value of C:
To find the value of the constant of integration, \(C\), we can substitute the given initial condition. It is mentioned that initially, the radius was 3.8 mm. This condition can be written as \(r(0) = 3.8\).
Substituting \(t = 0\) and \(r = 3.8\) in the radius function, we get:
\(3.8 = 2.1 \ln|1(0) + 5| + C\).
\(3.8 = 2.1 \ln(5) + C\).
Solving for \(C\):
\(C = 3.8 - 2.1 \ln(5)\).
Step 7: Substitute the value of C and evaluate for \(t = 27\):
Now, substitute the value of \(C\) into the radius function:
\(r = 2.1 \ln|1t + 5| + (3.8 - 2.1 \ln(5))\).
Finally, substitute \(t = 27\) into the expression for \(r\) and calculate the radius:
\(r = 2.1 \ln|1(27) + 5| + (3.8 - 2.1 \ln(5))\).
Evaluate this expression to find the radius after 27 seconds.