Questions: A superhero is rendered powerless when exposed to 45 or more grams of a certain element. A 450 year old rock that originally contained 400 grams of this element was recently stolen from a rock museum by the superhero's nemesis. The half-life of the element is known to be 150 years. a) How many grams of the element are still contained in the stolen rock? b) For how many years can this rock be used by the superhero's nemesis to render the superhero powerless? a) The stolen rock still contains about grams of the element. (Do not round until the final answer. Then round to two decimal places as needed.)

Solution Steps

The decay of the element in the rock can be modeled using the exponential decay formula:

\[ N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T}} \]

where:

• \( N(t) \) is the amount of the element remaining after time \( t \),
• \( N_0 \) is the initial amount of the element,
• \( T \) is the half-life of the element.

Given:

• Initial amount, \( N_0 = 400 \) grams,
• Half-life, \( T = 150 \) years,
• Time elapsed, \( t = 450 \) years.

Substitute these values into the decay formula:

\[ N(450) = 400 \left(\frac{1}{2}\right)^{\frac{450}{150}} \]

Simplify the exponent:

\[ N(450) = 400 \left(\frac{1}{2}\right)^3 \]

Calculate the remaining amount:

\[ N(450) = 400 \times \frac{1}{8} = 50 \text{ grams} \]

The superhero is rendered powerless when exposed to 45 or more grams of the element. We need to find the time \( t \) when the remaining amount is 45 grams.

Set up the equation:

\[ 45 = 400 \left(\frac{1}{2}\right)^{\frac{t}{150}} \]

Solve for \( t \):

\[ \left(\frac{1}{2}\right)^{\frac{t}{150}} = \frac{45}{400} = 0.1125 \]

Take the logarithm of both sides:

\[ \frac{t}{150} \log\left(\frac{1}{2}\right) = \log(0.1125) \]

Solve for \( t \):

\[ t = 150 \times \frac{\log(0.1125)}{\log(0.5)} \]

Calculate \( t \):

\[ t \approx 150 \times \frac{-0.9472}{-0.3010} \approx 472.1 \text{ years} \]

Final Answer