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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

Respuesta :

SaniShahbaz

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:

[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 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:

[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 = $ 525

Using 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.

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