Question

The interest rate r required to increase your investment p to the amount a in t years is found by
r=(a/p)1/t-1
. Find the interest rate r for p = 8100, a = 10,000, and t = 2. Round to the nearest hundredth.

I tried doing it and ended up getting a really weird answer. The choices are
A) .11%
B) 111.11%
C) 11.11%
D) 1.11%

Answer 1

When I tried to plug everything in, I ended up getting a negative number for some reason?

Answer 2

your formula is,

\(\large\color{black}{ \displaystyle r=\left(\frac{a}{p}\right)^{ \frac{1}{t}-1}}\)

is that right ?

Answer 3

Yeah

Answer 4

\(\large\color{black}{ \displaystyle r=\left(\frac{10{\tiny~~}000}{8{\tiny~~}100}\right)^{ \frac{1}{2}-1}}\) this is after plugging in the information

Answer 5

simplifying in the exponent
\(\large\color{black}{ \displaystyle r=\left(\frac{10{\tiny~~}000}{8{\tiny~~}100}\right)^{ -\frac{1}{2}}}\)

Answer 6

\(\large\color{black}{ \displaystyle r=\left(\frac{100^2}{90^2}\right)^{ -\frac{1}{2}}}\)

Answer 7

\(\large\color{black}{ \displaystyle r=\left(\left[\frac{100}{90}\right]^2\right)^{ -\frac{1}{2}}}\)

Answer 8

\(\large\color{black}{ \displaystyle r=\left(\frac{100}{90}\right)^{ 2\times \left(-\frac{1}{2}\right)}}\)

Answer 9

\(\large\color{black}{ \displaystyle r=\left(\frac{100}{90}\right)^{-1}}\)

Answer 10

can you solve from here ?

Answer 11

I think so... So it'd be A? I think? Technically, I got .9 and I'm not sure how that happened...

Answer 12

Is the formula
\(\large\color{black}{ \displaystyle\large\color{black}{ \displaystyle r=\left(\frac{a}{p}\right)^{ \frac{1}{t}-1}}}\)

or is it
\(\large\color{black}{ \displaystyle\large\color{black}{ \displaystyle r=\left(\frac{a}{p}\right)^{ \frac{1}{t-1}}}}\)

???

Answer 13

It's the first one

Answer 14

well, then the answer indeed is 0.9, i.e. 90%

Answer 15

with second one we would get something like
1.2345

Answer 16

so I guess you should verify if that is the correct formula.

Answer 17

maybe it is p/a ?

Answer 18

I can't do that until tomorrow /: I guess I'll just leave it until then and then check it.

I actually have another question I'm having trouble with... Can you try to help?

Answer 19

if the formula (in question 1) was p/a (instead of a/p) then you would be getting 1.11

Answer 20

I have to rationalize the denominator, which I vaguely know how to do, but only if there's just one number on the bottom. The equation is:
5 + sqrt 2 over 8 - sqrt 2

Answer 21

And it's definitely a/p... I know that part is right

Answer 22

\(\large\color{black}{ \displaystyle \displaystyle \frac{5+\sqrt{2}}{8-\sqrt{2}} }\)

Answer 23

Yes... Sorry, I don't know how to do that...

Answer 24

do you know a difference of squares (?).

\(\Large\color{black}{ \displaystyle ({\rm \color{blue}{b}}+{\rm \color{red}{a}})({\rm \color{blue}{b}}-{\rm \color{red}{a}})= }\)

\(\Large\color{black}{ \displaystyle {\rm \color{blue}{b}}({\rm \color{blue}{b}}-{\rm \color{red}{a}})+{\rm \color{red}{a}}({\rm \color{blue}{b}}-{\rm \color{red}{a}})=}\)

\(\Large\color{black}{ \displaystyle {\rm \color{blue}{b}}^2-{\rm \color{red}{a}}{\rm \color{blue}{b}}+{\rm \color{red}{a}}{\rm \color{blue}{b}}-{\rm \color{red}{a}}^2=}\)

\(\Large\color{black}{ \displaystyle {\rm \color{blue}{b}}^2~\cancel{-{\rm \color{red}{a}}{\rm \color{blue}{b}}}~\cancel{+{\rm \color{red}{a}}{\rm \color{blue}{b}}}-{\rm \color{red}{a}}^2=}\)

\(\Large\color{black}{ \displaystyle {\rm \color{blue}{b}}^2-{\rm \color{red}{a}}^2.}\)

Answer 25

I am just showing why \(\large\color{black}{ \displaystyle (a-b)(a+b)=a^2-b^2}\)
but do you know this rule (that I posted in black in this reply) ?

Answer 26

I think so... I think we went over it a few weeks ago. (I've been out of school a lot recently because of a family situation, so everything we've gone over is kind of hit and miss right now... I'm trying to catch up

Answer 27

yes,
and do you know that for any number 'a'

\(\large\color{black}{ \displaystyle \left(\sqrt{a}~\right)^2=a }\)

such as,
\(\large\color{black}{ \displaystyle \left(\sqrt{2}~\right)^2=2 }\)

Answer 28

Yeah

Answer 29

now back to our question
\(\large\color{black}{ \displaystyle \displaystyle \frac{5+\sqrt{2}}{8-\sqrt{2}} }\)

on our bottom, if we had

\(\large\color{black}{ \displaystyle \displaystyle (8~)^2-(\sqrt{2}~)^2 }\)

would that be good or bad ?

Answer 30

It'd be good, right? Because you could easily get the (sqrt 2)^2 to be just 2. Right?

Answer 31

yes, ( sqrt(2) )^2 would just be 2, and 8^2 is 64.
So when you subtract the bottom becomes a rational number

Answer 32

now, we need to figure out,
BY WHAT NUMBER TO MULTIPLY ON TOP AND BOTTOM
to make our expression

\(\large\color{black}{ \displaystyle \displaystyle \frac{5+\sqrt{2}}{8-\sqrt{2}} }\)

[to] have a desired denominator of

\(\large\color{black}{ \displaystyle \displaystyle \frac{{\rm \small ~disregard~~for~~now~}}{(8)^2-(\sqrt{2}~)^2} }\) ???

Answer 33

again the question says:
\(\large\color{black}{ \displaystyle \displaystyle \frac{5+\sqrt{2}}{8-\sqrt{2}} }\)
(just not to scroll back every time)



we know that

\(\Large\color{black}{ \displaystyle ({\rm \color{blue}{b}}+{\rm \color{red}{a}})({\rm \color{blue}{b}}-{\rm \color{red}{a}})={\rm \color{blue}{b}}^2-{\rm \color{red}{a}}^2 }\)


AND
\(\Large\color{black}{ \displaystyle( }\) you have the \(\Large\color{black}{ \displaystyle ({\rm \color{blue}{b}}-{\rm \color{red}{a}}) }\) part on the bottom \(\Large\color{black}{ \displaystyle) }\)

Answer 34

your b is 8, and a is sqrt(2)

Answer 35

So... I'm sorry, I'm kind of confused.
The denominator would end up being 62 after subtracting them, right? And then I'm not sure what to do at all...

Answer 36

I mean that you need

\(\large\color{black}{ \displaystyle \displaystyle \frac{(5+\sqrt{2})\color{blue}{\times (8+\sqrt{2}) }}{(8-\sqrt{2}) \color{blue}{\times (8+\sqrt{2}) }} }\)

Answer 37

see what I am doing ?

Answer 38

the denominator you have, it is 62

Answer 39

and for the numerator you will need to expand

Answer 40

Okay. So then to solve the top, would you basically use the FOIL method? I'm not sure if I'm getting this or not... I'm really confused

Answer 41

what did you get for the top