Question

A 22 gram sample of a substance that's used to sterilize surgical instruments has a k-value of 0.1363.
Formula: N=N0e^-kt
Find the substance's half-life, in days. Round to nearest tenth.
N0=initial mass (at time t=0)
N=mass at time t
k=a positive constant that depends on the substance itself and on the units used to measure time
t=time, in days

Answer 1

set \[\large e^{-0.1363}=0.5\] and solve for \(t\)

Answer 2

how...

Answer 3

half life means the time it takes to get to one half or \(0.5\)

Answer 4

takes two steps only

Answer 5

first write in equivalent logarthmic form as \[-0.1363t=\ln(0.5)\]

Answer 6

the to solve for \(t\) divide to get \[\large t=\frac{\ln(0.5)}{-0.1363}\]

Answer 7

and of course a calculator

Answer 8

can you walk me through all of that? I'm sorry I'm really struggling with this section in math..

Answer 9

ok although i already wrote the steps, i will add some explanation, maybe that will help

Answer 10

yes please

Answer 11

the inverses of the function \(f(x)=e^x\) is the natural log function \(\ln(x)\)

Answer 12

one thing you should be able to do is to switch from exponential form to logarithmic form quickly and easily
for example \[10^3=1000\] same as \[\log_{10}(1000)=3\]

Answer 13

\[2^4=16\\
\log_2(16)=4\] they same the same thing

Answer 14

would the answer be -80.70?

Answer 15

no the answer is not negative

Answer 16

so its 80.70

Answer 17

\[e^x=y\\
\ln(y)=x\]

Answer 18

\[e^{-0.1363t}=0.5\\
-0.1363t=\ln(.5)\]

Answer 19

hold on hold on what do i need to plug in this equation t=ln(0.5)/0.1363

Answer 20

yes

Answer 21

oh i see the mistake

Answer 22

what do i need to plug in?

Answer 23

the denominator should be negative

Answer 24

what?

Answer 25

lets go back to the start \[e^{-0.136t}=0.5\\
-0.1363t=\ln(0.5)\\
t=\frac{\ln(0.5)}{-0.1363}\]

Answer 26

ok ok what do i plug into this equation?

Answer 27

you have nothing to "plug in"
type in to your calculator (not sure which one you are using )
\[\ln(.5)\div (-.1363)\]

Answer 28

i use this
http://www.wolframalpha.com/input/?i=ln%28.5%29%2F-.1363

Answer 29

what is ln

Answer 30

5.09?

Answer 31

the natural log function , log base \(e\) \[\log_e(x)=\ln(x)\]

Answer 32

oooooh ok

Answer 33

so my answer is 5.08?

Answer 34

that is what i got, yes

Answer 35

final answer?

Answer 36

or correctly rounded \(5.09\)

Answer 37

ok thank you

Answer 38

oh wait

Answer 39

it asks "to the nearest tenth" so \(5.1\)

Answer 40

it said it was incorrect

Answer 41

oh man lol

Answer 42

oh well it'll buff thank you for teaching me. X

Answer 43

now i have a question

Answer 44

ok :)

Answer 45

what is your instagram?

Answer 46

lol give me yours

Answer 47

guess

Answer 48

satellite73?

Answer 49

bingo