MHB Find Half-Life of Morphine & Kool-Aid Powder in Blood/Water

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The half-life of morphine in the bloodstream is 3 hours, and it takes approximately 9 hours for the concentration to decrease from 0.4 mg to 0.01 mg. For the Kool-Aid powder problem, after 1 minute, 3 grams remain, and after 3 minutes, only 1 gram is left. The initial amount of Kool-Aid powder can be calculated using the decay pattern observed. Participants are encouraged to share their attempts to solve these problems for collaborative assistance.
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(a) The half-life of morphine in the bloodstream is 3 hours. Suppose that there's
initially 0.4 mg of morphine in the system, how long does it take until there's
only 0.01 mg of morphine remaining in the bloodstream?

(b) Suppose that there is initially x0 (0 is a subscript) grams of Kool-Aid powder in a glass of water.
After 1 minute there are 3 grams remaining and after 3 minutes there is only 1
gram remaining. Find x0 (0 is a subscript) and the amount of Kool-Aid powder remaining after 5 minutes.

Thanks in advance!
 
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Hi ayahouyee, (Wave)

Welcome to MHB!

We want to help you know how to do these problems on your own, so if we just do them for you that won't really be the best way to do this. We can work with you to solve your problems together.

So what have you tried? Show us where you first get stuck and we can help you keep going. :)
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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