I've been given a graph: Q equals 16e^-0.15t shown in the image below: A. What's the initial value of Q when t equals 0? so Q (0) = ? B. And this quantity decays at a continuous rate of what %?

Question

C. Now estimate the value of t when Q equals 4 using the graph below: so t≈

D. Now use logs to find exact value of t when Q = 4, so t = ?

Answer 1

Hi LPL L.,

Given Q(t) = 16e^-0.15t :

Q(0) = 16e^-0.15(0) = 16e⁰ = 16(1) = 16

For the decay rate percentage write Q(t) in the form of y = a(b)^x where a is the initial value, b is the growth/decay factor (for this problem - decay factor).

Q(0) = 16(e^-0.15)^t, where e^-0.15 is our b.

b = e^-0.15 = 0.8607.

b - 1 is the growth rate, and 1 - b is the decay rate.

The decay rate is 1 - 0.8601 = 0.1393, Written as a percentage, 0.1393 = 13.93%

For Q = 4 graphically, look at the vertical axis at 4, go horizontally from 4 to the exponential curve, then go vertically down to the horizontal axis to estimate t. I estimate t = 9.25.

Mathematically for Q(t) = 4:

Q(t) = 16e^-0.15t

4 = 16e^-0.15t, ... set Q(t) = 4

1/4 = e^-0.15t, ... divide both sides by 16

ln(.25) = -0.15t, ... take the natural log (ln) of both sides, ln(e^-0.15t) = -0.15t*ln(e) = -0.15t; (ln(e) = 1).

t = ln(.25)/(-0.15) = 9.24, ... divide both sides by -0.15t

I hope this helps, Joe.