Carbon-14 calculation

M

Matbatty-Challenged

New member
Joined
May 18, 2021
Messages
8
Hello,
I am a first-year college student studying Math and I am having a lot of difficulty trying to solve a problem from this week’s class:
  • 1622006528542.png

  • Many thanks for any help offered.
 
Matbatty-Challenged said:
Hello,
I am a first-year college student studying Math and I am having a lot of difficulty trying to solve a problem from this week’s class:
Click to expand...

  • You are given an equation relating N, No, k & t

  • You need to calculate 't'

  • You are given
    • N = 12​
    • No = 24​
    • k = 1.20968 * 10-4
      ​
    • calculate 't' - Exactly where are you stuck?
 
Last edited by a moderator:
You have a formula saying one variable DEPENDS on two other variables. A formula equates a dependent variable to an expression involving one or more independent variables. In your specific case, the formula tells you how to find a numeric value for N (the dependent variable) if you know the relevant numeric values for N0 and t (the independent variables). Could you calculate N if you knew N0 and t? It’s just plug and chug.

Most of our scientific knowledge is summarized in formulas.

It frequently turns out to be the case that we know the numeric value of the dependent variable and all but one of the independent variables and need to know the numeric value of that independent variable. This is a general problem of constant practical importance. The general solution is to perform mathematical operations on the formula to create a new formula where the unknown independent variable has become the dependent variable and the known dependent variable has become an independent variable. To see what I mean, let’s take this problem

[MATH]\text {formula where } N \text { is the dependent variable and the independent variables are } N_0 \text { and } t \implies \\ \text {formula where } t \text { is the dependent variable and the independent variables are } N_0 \text { and } N.[/MATH]The arrow here represents some mathematically valid process.

This use of and manipulation of formulas is one of the most important practical uses of mathematics. A lot of word problems will become clear if you realize that their solution depends on recognizing the relevant formula, manipulating it if necessary into a more relevant formula, and doing a plug and chug.

OK, you say, that is all well and good, but this general explanation does not help me solve this specific problem because the general explanation does not mention what the mathematically valid process is for any specific problem.

True. Whenever you see a formula or equation with a variable as an exponent and you need to manipulate it, a tool to think about is logarithms. Mathematically it makes no difference which system of logarithms you use, but picking the right base makes your work easier. Here choosing e as the base simplifies the math. WHY?

[MATH]N = N_0 e^{-kt} \implies ln(N) = ln(N_0e^{-kt}) = ln(N_0) + ln(e^{-kt}) [/MATH]
Can you finish up?
 
JeffM said:
You have a formula saying one variable DEPENDS on two other variables. A formula equates a dependent variable to an expression involving one or more independent variables. In your specific case, the formula tells you how to find a numeric value for N (the dependent variable) if you know the relevant numeric values for N0 and t (the independent variables). Could you calculate N if you knew N0 and t? It’s just plug and chug.

Most of our scientific knowledge is summarized in formulas.

It frequently turns out to be the case that we know the numeric value of the dependent variable and all but one of the independent variables and need to know the numeric value of that independent variable. This is a general problem of constant practical importance. The general solution is to perform mathematical operations on the formula to create a new formula where the unknown independent variable has become the dependent variable and the known dependent variable has become an independent variable. To see what I mean, let’s take this problem

[MATH]\text {formula where } N \text { is the dependent variable and the independent variables are } N_0 \text { and } t \implies \\ \text {formula where } t \text { is the dependent variable and the independent variables are } N_0 \text { and } N.[/MATH]The arrow here represents some mathematically valid process.

This use of and manipulation of formulas is one of the most important practical uses of mathematics. A lot of word problems will become clear if you realize that their solution depends on recognizing the relevant formula, manipulating it if necessary into a more relevant formula, and doing a plug and chug.

OK, you say, that is all well and good, but this general explanation does not help me solve this specific problem because the general explanation does not mention what the mathematically valid process is for any specific problem.

True. Whenever you see a formula or equation with a variable as an exponent and you need to manipulate it, a tool to think about is logarithms. Mathematically it makes no difference which system of logarithms you use, but picking the right base makes your work easier. Here choosing e as the base simplifies the math. WHY?

[MATH]N = N_0 e^{-kt} \implies ln(N) = ln(N_0e^{-kt}) = ln(N_0) + ln(e^{-kt}) [/MATH]
Can you finish up?
Click to expand...

Hello JeffM,

Thank you very much for such a detailed and careful explanation. With your help I was finally able to work out the problem. Much appreciated.

Regards,

M-C
 
You must log in or register to reply here.
Top