How Does Caffeine Decrease in Your Bloodstream Over Time?

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Caffeine levels in the bloodstream decrease over time, and the relationship is expressed by the equation t = 5(log P/ log 0.5). To find how long it takes for caffeine to drop to 25% of the original amount, one can substitute P with 25% in the equation. If coffee is consumed at 5:00 pm, by 10:30 pm, the remaining caffeine percentage can be calculated using the same formula. In this scenario, P is known as 25%, allowing for the calculation of t. Understanding this relationship helps in predicting caffeine levels at specific times.
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I'm having trouble with this question.

The percent "P", of caffeine remaining in your bloodstream is related to the elapsed time, "t', in hours, by t = 5(log P/ log 0.5)

a) How long will it take for the amount of caffeine to drop to 25% of the amount consumed?

b) Suppose you drink a cup of coffee after dinner at 5:00 pm. What % of the caffeine will remain in your system when you go to bed at 10:30 pm.
 
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a) What would P be if the caffeine in your blood stream is now 25% of the original amount consumed?

b) What variable, t or P, do you know in this case? How would you solve for the other one?
 

Thread 'Family of lines that are at a distance of 5 from the origin'

I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
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