To find the initial amount of krypton-81 present when the ice first formed, we can use the decay formula:
[tex]\[ N = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T}} \][/tex]
where:
- [tex]\( N \)[/tex] is the final amount of krypton-81 (1.675 grams in this case),
- [tex]\( N_0 \)[/tex] is the initial amount of krypton-81,
- [tex]\( t \)[/tex] is the age of the sample (458,000 years),
- [tex]\( T \)[/tex] is the half-life of krypton-81 (229,000 years).
To solve for [tex]\( N_0 \)[/tex], we need to rearrange the formula:
[tex]\[ N_0 = \frac{N}{\left(\frac{1}{2}\right)^{\frac{t}{T}}} \][/tex]
First, calculate the exponent:
[tex]\[ \frac{t}{T} = \frac{458,000}{229,000} = 2 \][/tex]
Next, calculate [tex]\( \left(\frac{1}{2}\right)^2 \)[/tex]:
[tex]\[ \left(\frac{1}{2}\right)^2 = \frac{1}{4} = 0.25 \][/tex]
Now, substitute [tex]\( N \)[/tex] and the exponent into the formula:
[tex]\[ N_0 = \frac{1.675}{0.25} = 6.7 \][/tex]
So, the initial amount of krypton-81 present when the ice first formed was 6.7 grams. Therefore, we fill the blank with:
The ice originally contained [tex]\(\boxed{6.7}\)[/tex] grams of krypton-81.
