Question

okay so i have a word problem that's exponential decay, but I cant figure out what to do.
2. A computer valued at $6500 depreciates at the rate of 14.3% per year.
a. Write a function that models the value of the computer.




b. Find the value of the computer after three years.



c. After how many years will the cost of the computer be $1800?

can someone help me?

Answer 1

Exponential equations are in the form of
y = a(b)^x

the initial value is going to be 'a'
we're decreasing by 14.3% a year

can you write 14.3% as a decimal?

the value of 'b' is going to be 1 - r
where r is that rate as a decimal

so what if 14.3% as a decimal

and then what is 1 minus that number

Answer 2

1.43 - 1 then? it'd be - 0.43?

Answer 3

no

14.3%

% means over 100

so what is 14.3 / 100

Answer 4

0.143?

Answer 5

yup

and then what is 1 - 0.143 = ?

that's the number that 'b' is going to be in y = a(b)^x

Answer 6

we subtract 0.143 from 1 because we're decreasing every year

if the value of the computer INCREASED each year by 14.3% then we would be ADDING

Answer 7

okay so 0.857 would be 1 - 0.143, the equation would be y = 6500(0.857)^x ?

Answer 8

Yup!

And then to answer part 2, you plug in x = 3
That will give you the value of the computer in 3 years

Answer 9

so it would be 4091.24? for the part 2 that is

Answer 10

Yes

Answer 11

Or actually, if you're rounding to the hundredths place then it would be 4091.25

Answer 12

For part 3, we want to figure out what 'x' is (that is, how many years it takes) when 'y' (or the price) is at 1800

The original equation is
y = 6500(0.857)^x

y is 1800

1800 = 6500(0.857)^x

To solve for 'x', first divide both sides by 6500

Answer 13

to*

Answer 14

so when there's multiple steps, you never ever want to round early

you want to keep as many digits until you reach the last step

so we have

0.276923 = (0.857)^x

do you know about the log rules?

\(\log a^b = b ~\log a\)

so if we take the log on both sides, we get

log 0.276923 = x * log 0.857

Answer 15

so now to solve for x, you just have to divide both sides by log 0.857

and you can first use a calculator and find out what log 0.276923 is equal to

and what log 0.857 is equal to

Answer 16

okay so log 0.276923 = -0.5576409722
and log 0.857 = -0.06701917808

so now i would just divide both sides by -0.06701917808?

Answer 17

yes, and what do you get then?

Answer 18

8.320617891 = x ?

Answer 19

you got it

Answer 20

so now it would be
1800 = 6500(0.857)^8.32 ?

Answer 21

uhh we just had to solve for x
8.32 is the answer

Answer 22

oh! okay, thank you so much!

Answer 23

You're welcome!