Math problem 13.1

Question

A computer purchased for $1,650 loses 11% 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) Give the function that models the decrease in value of the computer: v(t)=   

(B) In how many years will the computer be worth half its original value? Round answer to 1 decimal place.  years

Answer 1

v(t) = 1650e^-.11t

e^-.11t = b^t

-.11t = tlnb

lnb = -.11

b = e^-.11 = about .8958

v(t) = 1650(e^-.11)^t

or

v(t)= 1650(.8958)^t

time to decrease to half value

is when

1/2 =.5 = .8958^t

t = log base .8958 of .5

= ln.5/ln.8958

= about 6.299 years

= 6.3 years

but use a different method

that doesn't assume continuous decreases in value

year value

0 1650

1 1650- .11(1650)= .89(1650)

2 .89(1650) - .11(.89)(1650) = (.89^2)(1650)

3 (.89^3)(1650)

...

6 (.89^6)(1650)

t v(t) = 1650(.89)^t

for half value

1/2= .89^t= .5

t = log base .89 of log .5

= ln.5/ln.89

= about 5.948 years to decrease to half value

=about 5.9

you probably want these last answers

v(t)=1650(.89)^t

and

5.9 years