Question 1190364
the half life formula is:
.5 = (1 + r) ^ 5700
1 + r is the growth factor per year.
take the 5700th root of both sides of this equation to get:
.5 ^ (1/5700) = 1 + r
solve for 1 + r to get:
1 + r = .9998784026.
that number is rounded to the number of decimal digits displayed in the calculator.
to get the most accurate value when using that factor, use the internally stored value rather than the displayed value.
to confirm the value is correct, then:
1 * .9998784026 ^ 5700 = .4999999286, using the displayed value.
1 * .999878402... ^ 5700 = .5, using the stored value in the calculator,.


now that you know the rate per year, you can solve for the rest of the problem.


a) Suppose that 40% of the original carbon remains in a piece of pottery.
Estimate the age of the pottery.
b) Suppose that 75% of the original carbon remains in a piece of pottery.
Estimate the age of the pottery.
c) Suppose that 2% of the original carbon remains in a piece of pottery.
Estimate the age of the pottery.


start with .5 = x ^ y
x is equal to our stored value of .9998784026.
note that this is the displayed value of x.
the actual value of x is stored in the calculator and has more decimal digits than the displayed value.


we want to solve for y.
take the log of both sides of this equation to get:
log(.5) = y * log(x).
divide both sides of this equaiton by log(x) to get:
log(.5) / log(x) = y
solve for y to get:
y = 5700.
note that, since x = .9998784026, the equation is really:
log(.5) / log(.998784026) = y.
once again, you are using the stored value of x and not the displayed value of x, for greater accuracy.


this confirms that we can find the value of y using this formula.
the value of y is the number of years.


now to solve the rest of the problems.


a) Suppose that 40% of the original carbon remains in a piece of pottery.
Estimate the age of the pottery.


if 40% of the original carbon remains, then the formula becomes:
log(.4) / log(x) = y
solve for y to get:
y = 7534.990141.


b) Suppose that 75% of the original carbon remains in a piece of pottery.
Estimate the age of the pottery.


if 75% of the original carbon remains, then the formula becomes:
log(.75) / log(x) = y
solve for y to get:
y = 2365.713746.


c) Suppose that 2% of the original carbon remains in a piece of pottery.
Estimate the age of the pottery.


if 2% of the original carbon remains, then the formula becomes:
log(.02) / log(x) = y
solve for y to get:
y = 32169.98028


the equation can be graphed as shown below:


<img src = "http://theo.x10hosting.com/2022/020303.jpg" >


let me know if you have any questions.


theo