The value of a piece of equipment is declining exponentially

The value of a piece of equipment is declining exponentially according to the function V = f(t), where v equals the value of the equipment (in dollars) and t equals the age of the equipment in years. When the equipment was two years old, its value was $200,000. When it was five years old, its value was $120,000. Find the decay constant for the value of the equipment. Correct your answer to 4 decimal places. Find, to the nearest integer, the initial value of the equipment when it was new. Using the answers in part and above, write the exponential decay function V = f(t). Find the half-life of the equipment. Correct your answer to 2 decimal places.

Solution

V=f(t)=Ae^{-kt}

Here , k is the decay constant, k>0 , minus sign as value decreases with time

a.

V(2)=200000=Ae^{-2k}

V(5)=120000=Ae^{-5k}

Dividing the two equations gives

200/120=e^{3k}

5/3=e^{3k}

Solving for k gives

k~0.17

Thsi is the value of the decay constant

b)

V(2)=200000=Ae^{-2k}

Substituting here value of k we found gives

200000=Ae^{-0.34}

Hence, A~280990

This is hte value when it was new

c)

THe exponential dcay function ,V=f(t)=280990exp(-0.17t)

d) Half life is time in which value reduces to half on inital value

So, 280990/2=280990exp(-0.17t)

1/2=exp(-0.17t)

Solving for t gives

t~4.08 years