# Exponential rate question

#### MathsHelpPlz

##### New member
"A sick person takes a tablet of mass 500mg whose contents are gradually absorbed into the body. The mass not yet absorbed after t minutes is modelled by the formula 500(e^(-ct)) mg. After 30 minutes only 50mg remains unabsorbed.

a) Find the value of c
b) find the rate at which the contents of the tablet are being absorbed
i) when the person first takes the tablet,
ii) when half of the tablet is absorbed."

I've managed to find c but unsure how to go about the rest of the question. Also, I'm unsure when questions want me to differentiate to find rates as opposed to using the idea that "A quantity Q growing (or decaying) exponentially according the the law Q= ae^(ct) has a rate of growth (or decay equal to c (or -c) times its current value". Thank you for your time, any help will be greatly appreciated.

Edit: I found the derivative as f'(x)=-500c(e^(-ct)), and subbed in zero as x for b) i), but Im an order of magnitude too low at 4.9mg instead of 49.9mg.
Edit: I can find the right values but an order of magnitude too low.

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#### JeffM

##### Elite Member
"A sick person takes a tablet of mass 500mg whose contents are gradually absorbed into the body. The mass not yet absorbed after t minutes is modelled by the formula 500(e^(-ct)) mg. After 30 minutes only 50mg remains unabsorbed.

a) Find the value of c
b) find the rate at which the contents of the tablet are being absorbed
i) when the person first takes the tablet,
ii) when half of the tablet is absorbed."

I've managed to find c but unsure how to go about the rest of the question. Also, I'm unsure when questions want me to differentiate to find rates as opposed to using the idea that "A quantity Q growing (or decaying) exponentially according the the law Q= ae^(ct) has a rate of growth (or decay equal to c (or -c) times its current value". Thank you for your time, any help will be greatly appreciated.
$$\displaystyle Q = ae^{ct} \implies \dfrac{dQ}{dt} = ae^{ct} * c = cQ.$$

The rule is a result of differentiation.

#### MathsHelpPlz

##### New member
$$\displaystyle Q = ae^{ct} \implies \dfrac{dQ}{dt} = ae^{ct} * c = cQ.$$

The rule is a result of differentiation.
Ah ok, so would the rate be 'c times the current value' rather than c?

#### MathsHelpPlz

##### New member
Thanks for your explanation of the rule. I understand it now and have got the correct order of magnitude.

#### JeffM

##### Elite Member
Ah ok, so would the rate be 'c times the current value' rather than c?
It is a rate so your answer will $$\displaystyle cQ\ mg\ per\ minute$$.

c itself is just a dimensionless constant.

#### Subhotosh Khan

##### Super Moderator
Staff member
It is a rate so your answer will $$\displaystyle cQ\ mg\ per\ minute$$.

c itself is just a dimensionless constant.
Actually, 'ct' is dimensionless - because it is used as e[SUP]ct.[/SUP]

#### JeffM

##### Elite Member
Actually, 'ct' is dimensionless - because it is used as e[SUP]ct.[/SUP]
Right you are. Do I have to go to the corner?