Find the initial amount in the sample and the amount remaining after 100 years. Round your answers to the nearest gram as necessary.
Understand the Problem
The question is asking us to find two things: the initial amount of cesium-137 in grams, and the amount remaining after 100 years. We will use the given exponential function to solve for both values.
Answer
Initial amount: \( 266 \) grams; Amount after 100 years: \( 23 \) grams.
Answer for screen readers
Initial amount: ( 266 ) grams
Amount after 100 years: ( 23 ) grams
Steps to Solve
- Finding the Initial Amount
To find the initial amount of cesium-137, we need to evaluate the function ( A(t) ) at ( t = 0 ).
[ A(0) = 266 \left( \frac{1}{2} \right)^{0/30} ]
Calculating this:
[ A(0) = 266 \times 1 = 266 \text{ grams} ]
- Calculating the Amount After 100 Years
Next, we will find the amount remaining after 100 years by plugging ( t = 100 ) into the function.
[ A(100) = 266 \left( \frac{1}{2} \right)^{100/30} ]
Simplifying the exponent:
[ = 266 \left( \frac{1}{2} \right)^{3.33} ]
Calculating this exponent:
[ = 266 \times 0.088 \approx 23.41 \text{ grams} ]
Now, rounding to the nearest gram:
[ \text{Amount after 100 years} \approx 23 \text{ grams} ]
Initial amount: ( 266 ) grams
Amount after 100 years: ( 23 ) grams
More Information
The initial amount of cesium-137 is the value of the function when no time has passed, while the calculation after 100 years demonstrates radioactive decay, significantly reducing the amount due to its half-life.
Tips
- Forgetting to evaluate ( A(0) ) correctly for the initial amount.
- Miscalculating the exponent when determining the remaining amount after 100 years.
- Failing to round to the nearest gram.
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