To determine how much krypton-81 was present when the ice first formed, we can use the formula for radioactive decay:
[tex]\[ N = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T}} \][/tex]
Here:
- [tex]\( N \)[/tex] is the current amount of krypton-81, which is 1.675 grams.
- [tex]\( N_0 \)[/tex] is the initial amount of krypton-81 we need to find.
- [tex]\( t \)[/tex] is the age of the ice, which is 458,000 years.
- [tex]\( T \)[/tex] is the half-life of krypton-81, which is 229,000 years.
First, let's rewrite the formula to solve for [tex]\( N_0 \)[/tex]:
[tex]\[ N_0 = N \left( \frac{1}{2} \right)^{-\frac{t}{T}} \][/tex]
Substitute the given values into the equation:
[tex]\[ N_0 = 1.675 \left( \frac{1}{2} \right)^{-\frac{458,000}{229,000}} \][/tex]
Simplify the exponent:
[tex]\[ -\frac{458,000}{229,000} = -2 \][/tex]
So the formula becomes:
[tex]\[ N_0 = 1.675 \left( \frac{1}{2} \right)^{-2} \][/tex]
We know that:
[tex]\[ \left( \frac{1}{2} \right)^{-2} = 2^2 = 4 \][/tex]
Thus:
[tex]\[ N_0 = 1.675 \times 4 \][/tex]
[tex]\[ N_0 = 6.7 \][/tex]
Therefore, the ice originally contained [tex]\( 6.7 \)[/tex] grams of krypton-81.
So, the ice originally contained [tex]\(\boxed{6.7}\)[/tex] grams of krypton-81.
