Question

Amy's car was worth $9,000 at the beginning of 2000 and the value of the car decreased exponentially, decreasing by 23% each year. a. Write a function (f) that determines the value of Amy's car (in dollars) in terms of the number of years (t) since the beginning of 2000 (f(t) = 9000(frac{77}{100})^t) b. How many years after the beginning of 2000 was Amy's car worth $6,000? c. How many years after the beginning of 2000 was Amy's car worth $3,200?

          Amy's car was worth $9,000 at the beginning of 2000 and the value of the car decreased exponentially, decreasing by 23% each year.
a. Write a function (f) that determines the value of Amy's car (in dollars) in terms of the number of years (t) since the beginning of 2000
(f(t) = 9000(frac{77}{100})^t)
b. How many years after the beginning of 2000 was Amy's car worth $6,000?
c. How many years after the beginning of 2000 was Amy's car worth $3,200?

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$Amy's car was worth 9,000 at the beginning of 2000 and the value of the car decreased exponentially, decreasing by 23% each year. a. Write a function (f) that determines the value of Amy's car (in dollars) in terms of the number of years (t) since the beginning of 2000 (f(t) = 9000(frac77100)^t) b. How many years after the beginning of 2000 was Amy's car worth6,000? c. How many years after the beginning of 2000 was Amy's car worth 3,200?$

Added by Richard C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Jason Horton

03/17/2023

Step 1

We know that the value of Amy's car decreases by 23% each year. So, the remaining value each year is 100% - 23% = 77% or 0.77. Show more…

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Transcript

00:01 All right, for your question, we're looking for exponential decay, basically.

00:05 So i'm going to start with a template for exponential decay, which would be a of t equals p, our starting amount, times one minus our rate of decay, raised to the t power, t standing for the compounding periods here, basically.

00:26 So what we know is her car originally is worth 9 ,000.

00:32 That's your principal.

00:34 And we're going to multiply that to 1 minus 23 % or 0 .23 for each year.

00:45 And that works out to 9 ,000.

00:48 And i see your answer as 77 over 100, which makes sense.

00:55 And that is marked correct.

00:57 So we have that part done.

01:00 You have that correct.

01:01 I probably would have left that as a decimal.

01:03 Just be easier to work with.

01:05 So i'm going to put that as 0 .77.

01:09 And now let's work on part b.

01:11 Part b we want to know after how many years, how many years after the beginning of 2000 will our car be worth 6 ,000? so we just put 6 ,000 for the left side of the equation equals 9 ,000 times 0 .77 to the t power.

01:29 And we want to get the exponent and the base on a side by itself.

01:33 So i'm going to divide by 9 ,000.

01:39 And that would cancel this out.

01:42 We'd end up with 2 thirds.

01:44 I'm just going to leave it as a fraction.

01:45 2 thirds equals 0 .77 to the t power.

01:51 Now what we do, let me give myself some space here.

01:56 What we do is we log both sides to solve for an exponent position.

02:04 So i log both sides, and that allows us to move the exponent to the front of the logarithm...

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