# Math Help - differential equation

1. ## differential equation

If we assume that a person's weight depends on the energy consumed minus energy ued, one model is that weight changes in proportion to the difference, so:

dw/dt = k(C - 38.5w) where w(0) = w subscript 0

where w(t) kg is the weight at t days, C is the daily calorie intake, and we assume 38.5 calories per kg per day are used

(i) if you wish to maintain a constant weight of 82 kg, what should be your daily calorie intake C?

(ii) If you weigh 100 kg, and you want to lose 10 kg in a month, what should C be? (Assume k = 1.3 * 10^(-4) kg/calorie and a month of 30 days) Is this result healthy?

can someone solve these questions for me, they are on a past test paper which does not have solutions available.

2. Originally Posted by razorfever
If we assume that a person's weight depends on the energy consumed minus energy ued, one model is that weight changes in proportion to the difference, so:

dw/dt = k(C - 38.5w) where w(0) = w subscript 0

where w(t) kg is the weight at t days, C is the daily calorie intake, and we assume 38.5 calories per kg per day are used

(i) if you wish to maintain a constant weight of 82 kg, what should be your daily calorie intake C?

(ii) If you weigh 100 kg, and you want to lose 10 kg in a month, what should C be? (Assume k = 1.3 * 10^(-4) kg/calorie and a month of 30 days) Is this result healthy?

can someone solve these questions for me, they are on a past test paper which does not have solutions available.
(i) $\frac{dw}{dt} = 0$ ... solve for $C$

(ii) solve $\int \frac{dw}{C - 38.5w} = \int k \, dt$ with initial condition $w(0) = 100$.

after you find $w(t)$, set $w(30) = 90$ and solve for $C$.

3. can u help me out the same way with that kind of template for some of my other posts about differential equation word problems?
I have an exam coming up and my prof hasn't touched any word problems,
if you could provide templates, i can understand them better and hopefully pass the exam

http://www.mathhelpforum.com/math-he...tric-func.html

http://www.mathhelpforum.com/math-he...d-problem.html

http://www.mathhelpforum.com/math-he...-diver-de.html

http://www.mathhelpforum.com/math-he...ates-flow.html

http://www.mathhelpforum.com/math-he...72473-ivp.html

http://www.mathhelpforum.com/math-he...ntial-eqn.html

4. can you tell me what your final answer for C is?
i'm getting 102375.57

as for the constant in part (i) i'm getting (C - 3850)
and the expression for C in part (ii) i'm getting
[3465 - 3850e^(30k)] / [1 - e^(30k)]

is this correct
can someone show me the correct solution step by step?

5. $\int \frac{dw}{C - 38.5w} = \int k \, dt$

$\int \frac{-38.5}{C - 38.5w} \, dw = -38.5\int k \, dt$

$\ln|C - 38.5w| = -38.5kt + A$

$C - 38.5w = Be^{-38.5kt}$

$w(0) = 100$

$C - 3850 = B$

$C - 38.5w = (C - 3850)e^{-38.5kt}$

$C - 38.5w = Ce^{-38.5kt} - 3850e^{-38.5kt}$

$C - Ce^{-38.5kt} = 38.5w - 3850e^{-38.5kt}$

$C(1 - e^{-38.5kt}) = 38.5w - 3850e^{-38.5kt}$

$C = \frac{38.5w - 3850e^{-38.5kt}}{1 - e^{-38.5kt}}$

$w(30) = 90$ ... $C \approx 1090 \, cal$

6. now i'm getting C = 3515
how do you do that last part w(30) = 90
do u rearrange and make w the subject then put t=30 and solve for C right?
can u show me that last step explicitly??

7. Originally Posted by razorfever
now i'm getting C = 3515
how do you do that last part w(30) = 90
do u rearrange and make w the subject then put t=30 and solve for C right?
can u show me that last step explicitly??

since the task was to find C, I solved for C ... sub in 90 for w, 30 for t, and the given value for k (0.00013 , correct?)

8. yeah i just realized i was making a careless mistake
i wasn't putting E in my calculator , instead just getting the value of
(-38.5 * 0.00013 * 30) and this was giving me C as 3515 but I got it now

9. Originally Posted by razorfever
can u help me out the same way with that kind of template for some of my other posts about differential equation word problems?
I have an exam coming up and my prof hasn't touched any word problems,
if you could provide templates, i can understand them better and hopefully pass the exam

http://www.mathhelpforum.com/math-he...tric-func.html

http://www.mathhelpforum.com/math-he...d-problem.html

http://www.mathhelpforum.com/math-he...-diver-de.html

http://www.mathhelpforum.com/math-he...ates-flow.html

http://www.mathhelpforum.com/math-he...72473-ivp.html

http://www.mathhelpforum.com/math-he...ntial-eqn.html