30) The decay equation for radon- 222 gas is known to be $$ y=y_{0} e^{-.018 t} $$ with $$ t $$ in days. About how long will it take the amount of radon in a sealed sample of air to decay to $$ 90 \% $$ of its original value?

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1. Start with the decay equation for radon-222:  
   $$
   y = y_0 e^{-0.018t}
   $$

2. We want to find the time $$ t $$ when the amount of radon is $$ 90\% $$ of the original, so we set:  
   $$
   y = 0.9y_0
   $$

3. Substitute $$ 0.9y_0 $$ into the equation:  
   $$
   0.9y_0 = y_0 e^{-0.018t}
   $$

4. Divide both sides of the equation by $$ y_0 $$ (assuming $$ y_0 \neq 0 $$):  
   $$
   0.9 = e^{-0.018t}
   $$

5. Take the natural logarithm of both sides:  
   $$
   \ln(0.9) = \ln\left(e^{-0.018t}\right)
   $$
   Using the property $$ \ln(e^x) = x $$, we have:  
   $$
   \ln(0.9) = -0.018t
   $$

6. Solve for $$ t $$:  
   $$
   t = \frac{-\ln(0.9)}{0.018}
   $$

7. Substitute the value $$ \ln(0.9) \approx -0.10536 $$:  
   $$
   t \approx \frac{-(-0.10536)}{0.018} = \frac{0.10536}{0.018} \approx 5.8533
   $$

Thus, it takes approximately $$ 5.85 $$ days for the amount of radon to decay to $$ 90\% $$ of its original value.
It takes approximately 5.85 days for the radon to decay to 90% of its original amount.

The decay equation for radon- 222 gas is known to be with in days.
About how long will it take the amount of radon in a sealed sample of air to decay
to of its original value?

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The decay equation for radon- 222 gas is known to be with in days. About how long will it take the amount of radon in a sealed sample of air to decay to of its original value?