Question

In the course of an experiment you are measuring a quantity that is expected to change according to an exponential model. Your first measurement of the quantity gives a reading of 50 units, while three hours later the reading is 48. Construct the formula for the function that represents this quantity as a function of time.

Answer 1

evidently it is decreasing at a rate of 4% every 3 hours, so a snap answer would be
\(50\times (.94)^{\frac{t}{3}}\)

Answer 2

you are probably supposed to find a model that looks like
\[p(t)=p_0e^{kt}\] which means solving an exponential equation
we can do that too if you like, but the first answer is much snappier

Answer 3

i think we're supposed to represent it as a decay formula, so like the p(t) function you posted

Answer 4

ok then you want
\[p(t)=p_0e^{kt}\] we start with \(p_0=50\) because that is how much you have at the beginning, so what we need is \(k\)

Answer 5

you know when \(t=3\) you get 48, so you can write
\[48=50e^{k\times 3}\] and solve for \(k\)

Answer 6

how do we solve for k? logarithms?

Answer 7

\[48=50e^{3k}\] divide by 50 get
\[.96=e^{3k}\] then yes, you need to take the log

Answer 8

you get
\[\ln(.96)=3k\] and so
\[k=\frac{\ln(.96)}{3}\] whatever that is (need a calculator)
then you are done

Answer 9

okay, thanks so much