Question

this table shows the value of a computer as a function of time elapsed since the computer was manufactured.
time (t, in years) | value (v)
1 | $400.00
3 | $100.00
5 | $25.00

the value of the computer decreased exponentially throughout the five-year period. what was the average annual amount of decrease (in dollars per year) over the two-year interval from t = 2 to t = 4?
(note: give the answer as a positive value.)
enter your response here:
only 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ., -, and / are allowed in your answer.
mixed numbers are entered by adding a space after the whole number. spaces are only allowed between whole numbers and fractions.

Explanation:

Step1: Find exponential decay model

Exponential decay: $V(t) = V_0 e^{-kt}$. Use $t=1, V=400$; $t=3, V=100$.
$\frac{100}{400} = e^{-k(3-1)} \Rightarrow 0.25 = e^{-2k} \Rightarrow \ln(0.25) = -2k \Rightarrow k = -\frac{\ln(0.25)}{2} = \ln(2)$.
$V_0 = 400e^{k(1)} = 400e^{\ln2} = 800$. So $V(t)=800e^{-(\ln2)t}=800(0.5)^t$.

Step2: Calculate V(2) and V(4)

$V(2)=800(0.5)^2=200$; $V(4)=800(0.5)^4=50$.

Step3: Compute average annual decrease

Total decrease: $200-50=150$. Over 2 years: $\frac{150}{2}=75$.

Answer:

75