Respuesta :
Answer:
Sugar will dissolve to half of it's amount in 2.41 minutes.
Step-by-step explanation:
The given question is incomplete; here is the complete question.
When the sugar is dissolved in water, the amount A that remains undissolved after t minutes satisfies the differential equation
[tex]\frac{dA}{dt}=-kA, (k>0)[/tex]
If 25% of the sugar dissolves after 1 min, how long does it take for half of the sugar to dissolve.
The given differential equation is [tex]\frac{dA}{dt}=-kA[/tex]
In other form, dA = -kA.dt
We further integrate the equation,
[tex]\int\limits {dA}\,=\int{-kA}\, dt[/tex]
[tex]\int\limits{\frac{dA}{A} }\,=-k\int\, dt[/tex]
lnA = -kt + c
Or [tex]A=e^{c-kt}[/tex]
A = [tex]e^{c}.e^{-kt}[/tex] -----(1)
Here [tex]e^{c}[/tex] is a constant.
For t = 0,
[tex]A=e^{c}[/tex]
Let [tex]e^{c}=A_{0}[/tex]
Therefore, equation (1) will become
A = [tex]A_{0}e^{-kt}[/tex]
If sugar dissolves 25% in 1 minutes then undissolved sugar will be
100 - 25 = 75%
Now from the equation
[tex]0.75A_{0}=A_{0}e^{-k\times 1}[/tex]
[tex]e^{-k}=0.75[/tex]
By taking natural log on both the sides
[tex]ln(e^{-k})=ln(0.75)[/tex]
k = 0.2877
Now we have to calculate the time to dissolve half of the sugar, that means half the sugar will be undissolved.
Form the equation,
[tex]0.5A_{0}=A_{0}e^{-0.2877t}[/tex]
[tex]0.5=e^{-0.2877t}[/tex]
ln(0.5) = [tex]ln(e^{-0.2877t})[/tex]
0.6931 = 0.2877t
t = 2.41 minutes
