SOLUTION: Hello, I am currently doing work on some exponential and logarithmic problems. I am stuck on a word problem and don't understand how to even begin. Please help me solve the follo

Question 1187529: Hello, I am currently doing work on some exponential and logarithmic problems. I am stuck on a word problem and don't understand how to even begin. Please help me solve the following problem:
A computer purchased for $750 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) 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
Thank you!
-Diana
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
the computer is purchased for 750.
it loses 11% of its value each year.
the function you are using is:
v(t) = a * b^t
a would be the original amount = 750
b would be the rate that it is growing.
if it shrinking, than b will be less than 1.
you are given that it loses 11% of its value each year.
11% is equal to .11.
11 is the rate percent.
.11 is the rate.
rate percent / 100 = rate
rate * 100 = rate percent.
since it is losing 11% of its value then it is growing by a rate of (1 minus .11) = .89
the formula becomes v(t) = 750 * .89 ^ t

you want to know how many years it will take for it to lose half its value.
replace v(t) with 750/2 to get:
750/2 = 750 * .89 ^ t
if you divide both sides of that equation by 750, you will get:
(750/2)/750 = .89 ^ t
simplify that to get:
375/750 = .89 ^ t
simplify further to get:
.5 = .89^t

to solve for t, take the log of both sides of the equation to get:
log(.5) = log(.89^t)
by one of the laws of logs, log(.89^t) = t * log(.89)
equation becomes:
log(.5) = t * log(.89
solve for t to get:
t = log(.5) / log(.89)
use your calculator to get:
= 5.948034681.

confirm by replacing t in our original equation with that value to get:
375 = 750 * .89 ^ 5.948034681.
evaluate to get:
375 = 750 * .5 = 375.
this confirms the solution is correct.

your solution is that t = 5.948034681

here's a reference on logarithm arithmetic.

https://www.purplemath.com/modules/logrules.htm

there is a formula that i use that is more informative.
the formula is f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interest rate per time period
(1 + r) is the growth rate per time period
n is the number of time periods.

your formula was v(t) = a * b ^ t
your a corresponds to my p
your b corresponds to my growth rate.

if you had used my formula, then:
p = 750
r = -.11
1 + r = 1 - .11 = .89
f = 375

the formula would have become 375 = 750 * .89 ^ n
the growth rate is per year
n is in years.
your time periods are therefore in years.

you would have solve for n the same way.
375 = 750 * .89 ^ n becomes:
375/750 = .89 ^ n
you would then take the log of both sides of the equation to get:
log(375/750) = log(.89^n) which would become:
log(375/750) = n * log(.89) which would become:
n = log(375/750) / log(.89) which would get you the same answer.

hopefully this helps.
let me know if you have any questions.

theo

Answer by ikleyn(52772)   (Show Source): You can put this solution on YOUR website!
.

To see many other similar (and different) solved problems, look into the lesson
    - Problems on appreciated/depreciated values
in this site.

Learn the subject from there.

By the way, at the end of this lesson, you will find the set of links to other lessons on typical logarithmic and exponential
word problems: population growth, radioactive decay, carbon dating, bacteria growth, medication decay etc.

It is a nice place to learn all these subjects in one package, presented in compact and clear form ( ! )

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