Practice Half-life of Radium-226

puremath

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The half-life of radium-226 is 1620 years.

A. How much of 2 g sample remains after 100 years?

B. Find the time required for 80% of the 2 g sample to decay?
 

MarkFL

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After working the first half-life problem, you should be able to do this one.
 

puremath

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puremath

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After working the first half-life problem, you should be able to do this one.
Is the formula needed k = [ln(1/2)]/(half-life)?
 

MarkFL

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No, check the previous threads.
 

puremath

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No, check the previous threads.
I will target this problem later as I have all day to do math. No place to go. NYC is currently in lock down mode.
 

puremath

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After working the first half-life problem, you should be able to do this one.
For part B, must the first step be (0.80)(1.92 grams)?
 

MarkFL

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(a) \(\displaystyle A(100)=(2\text{ g})\left(\frac{1}{2}\right)^{\frac{100}{1620}}\approx1.91623115637718588\text{ g}\)

(b) \(\displaystyle \frac{1}{5}=\left(\frac{1}{2}\right)^{\frac{t}{1620}}\)

Solve for \(t\).
 
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puremath

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(a) \(\displaystyle A(100)=(2\text{ g})\left(\frac{1}{2}\right)^{\frac{100}{1620}}\approx1.91623115637718588\text{ g}\)

(b) \(\displaystyle \frac{1}{5}=\left(\frac{1}{2}\right)^{\frac{t}{1620}}\)

Solve for \(t\).
Please, explain your set up for B. I will solve for t as we will continue with exponential growth and decay throughout the remainder of the week in addition to starting trigonometry tomorrow.
 

puremath

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(a) \(\displaystyle A(100)=(2\text{ g})\left(\frac{1}{2}\right)^{\frac{100}{1620}}\approx1.91623115637718588\text{ g}\)

(b) \(\displaystyle \frac{1}{5}=\left(\frac{1}{2}\right)^{\frac{t}{1620}}\)

Solve for \(t\).
My answer for B makes no sense.

MathMagic200324_1.png
 

MarkFL

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\(\displaystyle 5=2^{\frac{t}{1620}}\)

\(\displaystyle t=1620\log_2(5)\approx3761.523513717527\quad\checkmark\)
 
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puremath

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\(\displaystyle 5=2^{\frac{t}{1620}}\)

\(\displaystyle t=1620\log_2(5)\approx3761.523513717527\quad\checkmark\)
Wow! I got it right. What does the value for t mean here?
 

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puremath

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I know that t is time but time is 3761.52?
 

MarkFL

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The unit of time here is years. With a half life of 1620 years, we know 75% will be gone in 2 half lives, or 3240 years, and 87.5% will be gone after 3 half-lives or 4860 years. So it seems reasonable that it would take some 3800 years for 80% to decay.
 
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puremath

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The unit of time here is years. With a half life of 1620 years, we know 75% will be gone in 2 half lives, or 3240 years, and 87.5% will be gone after 3 half-lives or 4860 years. So it seems reasonable that it would take some 3800 years for 80% to decay.
Wow! We will not be here for that.