differential equation problem help needed

bubblegum5678

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Assume that the dynamics of caffeine absorption is given by
ct + 1 = 0.87ct + d,
where t is time in hours and d is the amount of caffeine taken every hour.


If the initial amount of caffeine is
c0 = 800
mg and no new caffeine is consumed, estimate the time needed for 60% of the caffeine to be eliminated from the body. (Round your answer to the nearest integer.)
 
Assume that the dynamics of caffeine absorption is given by
ct + 1 = 0.87ct + d,
where t is time in hours and d is the amount of caffeine taken every hour.


If the initial amount of caffeine is
c0 = 800
mg and no new caffeine is consumed, estimate the time needed for 60% of the caffeine to be eliminated from the body. (Round your answer to the nearest integer.)
There's something wrong here. The equation says that d would have to be a function of t, not a constant. Please check your problem statement.

And this should be a differential equation...

-Dan
 
I'm going to guess that you do NOT mean "ct+ 1", 1 added to the number ct, but rather "ct+1\displaystyle c_{t+1}" and that this is NOT a "differential equation" but rather a "difference equation": ct+1=0.86ct+d\displaystyle c_{t+1}= 0.86c_t+ d. But we are also told that d= 0 so this is just ct+1=0.86ct\displaystyle c_{t+1}= 0.86c_t. That is, the amount of caffeine in the body at each hour is 0.86 times the amount of caffeine the previous hour. If c0\displaystyle c_0 is the initial amount of caffeine in the body then the amount after one hour is c1=0.86c0\displaystyle c_1= 0.86c_0, after 2 hours, c2=0.86c1=0.86(0.86c0)=(0.86)2c0\displaystyle c_2= 0.86c_1= 0.86(0.86c_0)= (0.86)^2c_0, after 3 hours, c3=0.86c2=(0.86)3c0\displaystyle c_3= 0.86c_2= (0.86)^3c_0, etc. In general, after n hours, cn=(0.86)nc0\displaystyle c_n= (0.86)^nc_0.

60% of the caffeine will have been eliminated when 40% is left: You want to find n such that (0.86)n=.40\displaystyle (0.86)^n= .40. Take the logarithm of both sides to find n.
 
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