Gary has 1400 to invest with the goal of having 4000...

Name: gary has 1400 to invest with the goal of having 4000 available to purchase a used car if he can earn 12 compounded semiannually on his investment how long will he have to wait to acquire his 19848
Uploaded: 2022-07-18T20:27:04-08:00
Duration: 2 min 48 s
Channel: Tim Thornhill
Description: gary has 1400 to invest with the goal of having 4000 available to purchase a used car if he can earn 12 compounded semiannually on his investment how long will he have to wait to acquire his 19848

Gary has $1,400 to invest with the goal of having $4,000 available to purchase a used car. If he can earn 12% compounded semiannually on his investment, how long will he have to wait to acquire his car? Question 16Select one: a. Eighteen months b. Nine years c. Eighteen years d. Nine months e. Four years

Transcript

00:01 This problem asks if danica has $1 ,200 to invest at 8 % per year compounded monthly, how long will it be before he has $2 ,400 if compounding is continuous, how long will it be? so we have the two formulas that we need listed here.

00:13 The first we'll use for compounded monthly, so he wants to know how long it will take for the y value, the final amount, to be $2 ,400, equal to the initial amount, which is $1 ,000, times 1 plus the rate, which was 8%, and it's compounded monthly, so our end value is 12, raised to the 12 value for n again times t so t is what we're trying to solve for so it's a solve for a variable in exponent we have to isolate the base so this one plus 0 .08 divided by 12 we need to get by itself so we'll divide 1200 over and 1200 divided into 2 so we're left with 2 equal to the 1 plus 0 .08 divided by 12 race to the 12 t now we'll use logarithms and i'm going to use natural log and that gives us the ability to take the exponent right as a coefficient.

01:00 So i'm going to do natural log of both sides and that's going to let us move this 12t in front of this natural log.

01:06 So 12t times natural log of 1 plus 0 .08 divided by 12 and that's equal to natural log of 2.

01:14 So if we're solving for t, we're going to divide by the entire 12 natural log 1 plus 0 .08 divided by 12 on both sides.

01:22 So natural log of 12 divided by the same thing.

01:24 12 times natural log 1 plus 0 .08 divided by 12.

01:29 And that's going to give us our t value.

01:31 And our t value for the first part of the problem is 8 .693.

01:38 So looking at our choices, it looks like b is the only choice that could be correct.

01:42 But we'll make sure that this is correct by working out the second part of the problem too.

01:46 The second part says if it's compounded continuously, how long will it take to reach the 2400? so same thing...

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