Question 1079291
f = p * (1 + r) ^ n


f is the future value
p is the present value
r is the interest rate per time period.
n is the number of time periods.


the quantity of medicine is decreasing by 25% every 40 minutes.


your time period is therefore every 40 minutes.


your rate is -25% every 40 minutes.


in the formula, you use the decimal equivalent of the percent which is equal to the percent / 100.


the interest rate per time period is therefore -.25


the formula of f = p * (1 + r) ^ n becomes


f = p * (1 - .25) ^ n which becomes:


f = p * .75 ^ n


in order for the value of the medicine to be equal to 1/2 what it was, the formula becomes:


1/2 = .75^n 


this is because f is 1/2 and p is 1, and 1 * .75 ^ n is the same as .75 ^ n.


you would solve for n using logs.


take the log of both sides of the eqution to get log(1/2) = log(.75^n)


because log(b^n) = n*log(b), this becomes:


log(1/2) = n*log(.75)


divide both sides of this equation by log(.75) and you get:


log(1/2) / log(.75) = n


solve for n to get n = 2.40942084


this means that that the original amount will be half of what it was in 2.40942084 time periods that are 40 minutes each.


translate this to minutes, and it becomes n = 2.40942084 * 40 = 96.37683359 minutes.


this formula will work regardless of the value of the milligrams in the blood stream.


start with 300 mg


you will have 150 mg in the blood stream in 2.40942084 time periods.


f = 300 * .75 ^ 2.40942084 = 150


f = 150 * .75 ^ 2.40942084 = 75


f = 75 * .75 ^ 2.40942084 = 37.5


etc.


the amount of drug in the bloodstream will be half of what it was every 2.40942084 time periods.


your original table can be set up in 40 minute intervals.


when n = 0, the value in this table will be 300 * .75 ^ 0 = 300.


when n = 1, the value in this table will be 300 * .75 ^ 1 = 225


when n = 2, the value in this table will be 300 * .75 ^ 2 = 168.75


when n = 3, the value in this table will be 300 * .75 ^ 3 = 126.5625


etc.


for the remaining dose being 2% of the original dose, you would calculate the time required in 40 minute intervals as follows:


2% of 300 = .02 * 300 = 6 mg.


in the formula of f = p * .75 ^ n, you would get:


6 = 300 * .75 ^ n


you will want to solve for n in this equation.


divide both sides of the equation by 300 to get:


6/300 = .75 ^ n


take the log of both sides of this equation to get:


log(6/300) = log(.75^n)


this becomes log(6/300) = n * log(.75)


divide both sides of the equation by log(.75) to get:


log(6/300) / log(.75) = n


solve for n to get n = 13.59842472


the original dose of 300 mg should reduce to 6 mg in 13.59842472 time periods that are 40 minutes each.


that's equivalent to 13.59842472 * 40 = 543.9369888 minutes which is equivalent to 543.9369888 / 60 = 9.06561648 hours.


to be on the safe side, you would round down so there is at least 2% of the medicine in the bloodstream.


to graph the equation, replace f with y and replace n with x to get:


y = 300 * .75^x


your graph will look  like this:


<img src = "http://theo.x10hosting.com/2017/050311.jpg" alt="$$$" </>


if your want your time intervals to be minutes instead of 40 minutes, you would do the following.


.75 ^ 1/40 = .9928337491


your growth rate per minute would be .9928337491 per 1 minute interval.


your time interval is now in minutes.


i will graph the equation of y = 300 * .9928337491 ^ x to show you that this is equivalent to the same equation when the time period is in 40 minute intervals.


your graph looks like this.


<img src = "http://theo.x10hosting.com/2017/050312.jpg" alt="$$$" </>


the value of x is now every minute rather than every 40 minutes.


you can also translate to hours.


your do that as follows:


hours relative to 40 minutes are 60/40 = 6/4 = 3/2


.75 ^ (3/2) = .6495190528


your equation becomes y = 300 * .6495190528 ^ x


the value of x is now every hour instead of every 40 minute interval.


the graph of that equation looks like this:


<img src = "http://theo.x10hosting.com/2017/050313.jpg" alt="$$$" </>


if you want to sketch a 4 hour graph, it would look like this:


<img src = "http://theo.x10hosting.com/2017/050314.jpg" alt="$$$" </>