Question

The table below compares the radioactive decay rates of two materials.

Based on the table, which of these conclusions is most likely correct?
The half-life of Material 1 and Material 2 are equal.
The half-life of Material 2 is double the half-life of Material 1.
The half-life of Material 2 is 30 hours more than the half-life of Material 1.
The half-life of Material 1 is 30 hours more than the half-life of Material 2.

Answer 1

@snowflake0531 wanna give it a try lol

Answer 2

ok so like a first step do you now what mean the half life of a material ?

Answer 3

i dont :(

Answer 4

but you need to know to solve this your problem

@supie any idea ?

Answer 5

@AZ can you explain here please - ? ty

Answer 6

any idea @AZ ? how you explain what is a half-life of a material ? hope you know what mean

Answer 7

we need to calculate the half life

Answer 8

so for material A

final amount = 13
initial amount = 208
time = 60 hours
half-life = x

we're calculating half life

can you plug it into the equation and solve?

we will have to use ln on both sides so we can solve for 'x'

Answer 9

and then you similarly find the half life of the second material

and then compare the half-lives

Answer 10

half-life of a material is the amount of time it takes for a material to become half

so if you had 100 grams of something
the amount of time it takes for that 100 grams to decay to 50 grams is called the half life

and that's how much time it would take for it to go from 50 grams to 25 grams
and from 25 grams to 12.5 grams

Answer 11

\(\color{#0cbb34}{\text{Originally Posted by}}\) @AZ
half-life of a material is the amount of time it takes for a material to become half

so if you had 100 grams of something
the amount of time it takes for that 100 grams to decay to 50 grams is called the half life

and that's how much time it would take for it to go from 50 grams to 25 grams
and from 25 grams to 12.5 grams
\(\color{#0cbb34}{\text{End of Quote}}\)
ty @AZ perfect explained

Answer 12

i got 13=208 (1/2) 60/x... what do i do from here?

Answer 13

divide 208 on both sides

Answer 14

what is 13/208 =

Answer 15

.0625

Answer 16

good so now we have

\( 0.0625 = \left(\dfrac{1}{2}\right)^{\dfrac{60}{x}}\)

if we take the natural log on both sides, we can bring down the exponent because

\( ln(a^b) = b ~ln(a)\)

Answer 17

so if we take the natural log on both sides, we get

\( ln (0.0625 )= \dfrac{60}{x} ln\left(\dfrac{1}{2}\right)\)

Answer 18

can you use a calculator and do

ln(0.0625) divided by ln(0.5)

Answer 19

we want to get x all by itself

Answer 20

.125?

Answer 21

did you understand what I did?

Answer 22

with the whole ln thing

If they're teaching you it, I'm sure they expect you to know it

Answer 23

\(\color{#0cbb34}{\text{Originally Posted by}}\) @artlover03
.125?
\(\color{#0cbb34}{\text{End of Quote}}\)

no
https://www.google.com/search?q=ln(0.0625)+%2F+ln(0.5)

Answer 24

oh.. 4

Answer 25

yes so we divided by ln(1/2) or ln(0.5) on both sides so now we get

4 = 60/x

can you solve for x?

that's going to be the half life of material 1

Answer 26

.0666 repeating?

Answer 27

no if you divide 60 by 0.0666 do you get 4?

so we have

4 = 60/x

so multiply x on both sides and you'll get

4x = 60

to get x all by itself, you need to divide 4 on both sides

Answer 28

your silence worries me

4x = 60

so x = 60/4

60/4 = ??

Answer 29

15

Answer 30

so that's the half life for the first material

we need to do it all over again for the second material

so for material 2

final amount = 50
initial amount = 200
time = 60 hours
half-life = x

\( 50 = 200\left(\dfrac{1}{2}\right)^{\dfrac{60}{x}}\)

Answer 31

so first things first, divide by 200 on both sides

Answer 32

.25?

Answer 33

good

and what do you get when take ln of both sides?

can you write the equation out

Answer 34

.25=200 (1/2) ^ 60/x

Answer 35

no, there would be no 200 because we divided the 200 on both sides

Answer 36

0.25 = (1/2) ^ (60/x)

Answer 37

1/2 is the same thing as 0.5 so let's re-write it

0.25 = (0.5)^ (60/x)

Answer 38

can you take the natural log on both sides?

Answer 39

\(\color{#0cbb34}{\text{Originally Posted by}}\) @AZ
good so now we have

\( 0.0625 = \left(\dfrac{1}{2}\right)^{\dfrac{60}{x}}\)

if we take the natural log on both sides, we can bring down the exponent because

\( ln(a^b) = b ~ln(a)\)
\(\color{#0cbb34}{\text{End of Quote}}\)
\(\color{#0cbb34}{\text{Originally Posted by}}\) @AZ
so if we take the natural log on both sides, we get

\( ln (0.0625 )= \dfrac{60}{x} ln\left(\dfrac{1}{2}\right)\)
\(\color{#0cbb34}{\text{End of Quote}}\)


remember this?

Answer 40

.5=60x?

Answer 41

uhhh no

Answer 42

we have

\( 0.25 = 0.5^{\left(\dfrac{60}{x}\right)}\)

Answer 43

if we take ln on both sides we get

\( ln(0.25) = \dfrac{60}{x}~ ln(0.5)\)

Answer 44

you should be able to solve for x

Answer 45

Follow all the steps I've already told you before.

Answer 46

umh.. 60/x (.5)= 30x?

Answer 47

i know im hard to teach, sorry

Answer 48

@AZ I THINK I GOT IT! I GOT 4 FOR THE SECOND PART.. SO IS THE ANSWER A) THEY ARE BOTH THE SAME?!?!

Answer 49

@artlover03 sorry, I had to step away- but no they are not both the same

what is ln(0.25) divided by ln(0.5) =

ln(0.25) / ln(0.5) =

just use a calculator
https://www.google.com/search?q=ln(0.25)+%2F+ln(0.5)+%3D

and then you know that number = 60/x

so just divide 60 by that number and tat's the half life of the second material