A computer purchased for $1,050 loses 19% of its value every year.The computer's value can be modeled by the function v(t)=a⋅b^t, where v is the dollar value and t the number of years since purchase.(A) In the exponential model a=____ and b=_____ .(B) In how many years will the computer be worth half its original value? Round answer to 1 decimal place.The answer is_____ years

Accepted Solution

Answer:A) a = 1050 and b = 0.81B) 3.3Step-by-step explanation:Original price of the computer = $ 1050Rate 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:[tex]v(t)=a(b)^{t}[/tex]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 1050b 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 0.81So, a = 1050 and b = 0.81The exponential model would be:[tex]v(t)=1050(0.81)^{t}[/tex]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 = $ 525Using the exponential model, we get:[tex]525=1050(0.81)^{t}\\\\ 0.5=(0.81)^{t}\\[/tex]Taking log of both sides:[tex]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[/tex]Thus, after 3.3 years the worth of computer will be half of its original price.