Math Help

[Yourmother]

Hello, would really appreciate some confirmation on the answers to these two questions. Calc 3 is giving me a lot of trouble, especially with separable equations.

1) A sample of tritium-3 decayed to 94.5% of its original amount after a year.

a) What is the half-life of tritium-3? b) How long would it take for the sample to decay to 20% of its original amount?
2) A cake is removed from an oven at 210 degree Fahrenheit and left to cool at room temperature, which is 70 degrees Fahrenheit. After 30 minutes the temperature of the cake is 140 degrees Fahrenheit. When will it be 100 degrees Fahrenheit?

 

[Tannhauser]

Dude, haven't you heard of the famous Math Thread? http://intpforum.com/showthread.php?t=23402

Anyway:

1 a) If we take the unit of time to be years, then we solve (1/2)^(-t/h) = 0.945 for h, where t is number of years. With t=1 this gives h = -1/log2(0.945), which is about 12.25 years.

1 b) Here you just put h into (1/2)^(-t/h) = 0.2 and solve for t, which gives about 28.45 years

2) I guess here you have to use Newton's law of cooling? That amounts to solving the differential equation

where y(t) is the temperature of the cake, r is some constant signifying the rate of cooling and T_e is the temperature of the room. The general solution (it's a first-order linear ODE) is

for some constant C, which is determined from the initial conditions: y(0)=210 and T_e = 70, so C=140. Do determine r we use the other piece of info, i.e.

So to get the time for y(t) = 100, it's the same thing as in 1b); solve for t

Feel free to ask for any clarifications

 

[Yourmother]

Woah Ok. Thanks so much man! Thumbs up for that!

 

[DeadChannel]

 

[Tannhauser]

^ that's quite accurate, actually.