Question

You deposit $5000 into an account that earns 5% compounded annually. A friend deposits $4750 into an account that earns 4.95% annual interest, compounded continuously. Will your friend's balance ever equal yours?

Answer 1

I tried making the equations for this:
yours:\[A=5000(1+.05)^t\]
friends: \[q=4750*e ^{0.0495*t}\]

Answer 2

and graphed them to see when they would intersect but found they didn't but my homework says that's wrong. Not sure where to go with this now

Answer 3

@ganeshie8 are you able to help with this one?

Answer 4

see if \(e^{0.0495} \gt 1+0.05\) is true

Answer 5

if it is, then the `continuous compounding` curve grows at a greater rate and overtakes the `annual compounding` curve at some point for sure

Answer 6

well unless my equations are wrong they ran pretty consistently alongside each other when I graphed it

Answer 7

graphing is a good idea,
but consider the possibility that they might cross much far away... half way before infinity ?

Answer 8

if the `continuous` growth rate is even a hair greater than the `annual compounding` growth rate, then the `continuous` wins in the long run

Answer 9

not with the curves they were at. they would have to curve back around towards each other which isn't a quality of exponential equations. so I did something wrong somewher

Answer 10

Did u check if below is true :

\(e^{0.0495} \gt 1+0.05\)

?

Answer 11

yeah, it's > by like o.ooo7

Answer 12

lol thats more than a hair

Answer 13

so ur friend's balance is growing at a slightly greater rate than your balance

Answer 14

so he will over take u after few years

Answer 15

you may not be living to see it though... he reaches it in approximately 72 years :
http://www.wolframalpha.com/input/?i=solve+5000%281%2B.05%29%5Et+%3D+4750*e+%5E%7B0.0495*t%7D

Answer 16

so is 72 years when our accounts will be equal? I'm confused

Answer 17

yup thats what wolfram says

Answer 18

but u dont need to work that

Answer 19

the question is asking u only whether he will ever reach ur balance or not

Answer 20

for that checking the growth rate is sufficient

Answer 21

no the question is asking WHEN he will equal mine

Answer 22

you better read the question again :/

Answer 23

or i must have overlooked something, let me go thru again lol

Answer 24

that's what the question says but the answer box says " ____years (if he never reaches your balance enter NEVER)"

Answer 25

72 was right. i just have no idea how you got it

Answer 26

can you just show me how you got 72?

Answer 27

ohhk.. cool :)

Answer 28

you wanto knw when the balances in both accounts equal,
so simply equate both the equations and solve for \(t\)

Answer 29

you equation :
\(A=5000(1+.05)^t \)

your friend's equation :
\(q=4750*e ^{0.0495*t}\)

Answer 30

set them equal :

\(5000(1+.05)^t=4750*e ^{0.0495*t}\)

Answer 31

solve

Answer 32

oh okay, that makes more sense, thank you :D

Answer 33

np :)

Answer 34

wait, one more question. How would you go about solving that? Would you take the log of both sides? your link to wolfram was kinda confusing on how to solve it. I have to show my work or I don't get credit

Answer 35

\(5000(1+.05)^t=4750*e ^{0.0495*t}\)


divide 4750 both sides :
\(\dfrac{5000}{4750}(1.05)^t=e ^{0.0495*t}\)


divide (1.05)^t both sides :
\(\dfrac{5000}{4750}= \dfrac{e ^{0.0495*t}}{(1.05)^t}\)

which is same as :
\(\dfrac{5000}{4750}= \left(\dfrac{e ^{0.0495}}{1.05}\right)^t\)

Answer 36

Now you can take log both sides and isolate \(t\)

Answer 37

Okay, i got it now.

Answer 38

Thank you :)

Answer 39

u wlc :)