Answer:
A) a = 1050 and b = 0.81
B) 3.3
Step-by-step explanation:
Original price of the computer = $ 1050
Rate of decrease in price = r = 19%
This means, every year the price of the computer will be 19% lesser than the previous year. In other words we can say that after a year, the price of the computer will be 81% of the price of the previous year.
Part A)
The exponential model is:
![v(t)=a(b)^{t}](https://tex.z-dn.net/?f=v%28t%29%3Da%28b%29%5E%7Bt%7D)
Here, a indicates the original price of the computer i.e. the price at time t = 0. So for the given case the value of a will be 1050
b represents the multiplicative rate of change i.e. the percentage that would be multiplied to the price of previous year to get the new price. For this case b would be 81% or 0.81
So, a = 1050 and b = 0.81
The exponential model would be:
![v(t)=1050(0.81)^{t}](https://tex.z-dn.net/?f=v%28t%29%3D1050%280.81%29%5E%7Bt%7D)
Part B)
We have to find after how many years, the worth of the computer will be reduced to half. This means we have the value of v which is 1050/2 = $ 525
Using the exponential model, we get:
![525=1050(0.81)^{t}\\\\ 0.5=(0.81)^{t}\\](https://tex.z-dn.net/?f=525%3D1050%280.81%29%5E%7Bt%7D%5C%5C%5C%5C%200.5%3D%280.81%29%5E%7Bt%7D%5C%5C)
Taking log of both sides:
![log(0.5)=log(0.81)^{t}\\\\ log(0.5)=t \times log(0.81)\\\\ t = \frac{log(0.5)}{log(0.81)}\\\\ t = 3.3](https://tex.z-dn.net/?f=log%280.5%29%3Dlog%280.81%29%5E%7Bt%7D%5C%5C%5C%5C%20log%280.5%29%3Dt%20%5Ctimes%20log%280.81%29%5C%5C%5C%5C%20t%20%3D%20%5Cfrac%7Blog%280.5%29%7D%7Blog%280.81%29%7D%5C%5C%5C%5C%20t%20%3D%203.3)
Thus, after 3.3 years the worth of computer will be half of its original price.