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Answer :
To solve this problem, we are using the formula for radioactive decay:
[tex]\[ A = A_0 e^{-0.00693x} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of radioactive material remaining after [tex]\( x \)[/tex] years.
- [tex]\( A_0 \)[/tex] is the initial amount of radioactive material.
- [tex]\( x \)[/tex] is the time in years.
- [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
In this problem, we are given:
- The initial amount [tex]\( A_0 = 500 \)[/tex] pounds.
- The decay rate constant is 0.00693.
- The time [tex]\( x = 70 \)[/tex] years.
Now, we need to find the amount remaining, [tex]\( A \)[/tex], after 70 years.
Step-by-step solution:
1. Start with the initial amount [tex]\( A_0 = 500 \)[/tex] pounds.
2. Substitute [tex]\( x = 70 \)[/tex] years and the decay rate constant into the formula:
[tex]\[ A = 500 \times e^{-0.00693 \times 70} \][/tex]
3. Calculate the exponent:
[tex]\[ -0.00693 \times 70 = -0.4851 \][/tex]
4. Calculate [tex]\( e^{-0.4851} \)[/tex]. This involves using a scientific calculator or mathematical software to find the value of [tex]\( e\)[/tex] raised to the power of [tex]\(-0.4851\)[/tex].
5. Multiply the initial amount by the calculated decay factor:
[tex]\[ A = 500 \times e^{-0.4851} \][/tex]
After performing these calculations, you will find:
[tex]\[ A \approx 307.82 \][/tex] pounds
Therefore, the amount of radioactive material remaining after 70 years is approximately 308 pounds.
The closest answer choice is:
b) 308 lb
[tex]\[ A = A_0 e^{-0.00693x} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of radioactive material remaining after [tex]\( x \)[/tex] years.
- [tex]\( A_0 \)[/tex] is the initial amount of radioactive material.
- [tex]\( x \)[/tex] is the time in years.
- [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
In this problem, we are given:
- The initial amount [tex]\( A_0 = 500 \)[/tex] pounds.
- The decay rate constant is 0.00693.
- The time [tex]\( x = 70 \)[/tex] years.
Now, we need to find the amount remaining, [tex]\( A \)[/tex], after 70 years.
Step-by-step solution:
1. Start with the initial amount [tex]\( A_0 = 500 \)[/tex] pounds.
2. Substitute [tex]\( x = 70 \)[/tex] years and the decay rate constant into the formula:
[tex]\[ A = 500 \times e^{-0.00693 \times 70} \][/tex]
3. Calculate the exponent:
[tex]\[ -0.00693 \times 70 = -0.4851 \][/tex]
4. Calculate [tex]\( e^{-0.4851} \)[/tex]. This involves using a scientific calculator or mathematical software to find the value of [tex]\( e\)[/tex] raised to the power of [tex]\(-0.4851\)[/tex].
5. Multiply the initial amount by the calculated decay factor:
[tex]\[ A = 500 \times e^{-0.4851} \][/tex]
After performing these calculations, you will find:
[tex]\[ A \approx 307.82 \][/tex] pounds
Therefore, the amount of radioactive material remaining after 70 years is approximately 308 pounds.
The closest answer choice is:
b) 308 lb
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