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Answer :
To solve this problem, we need to determine how much gold-198 ([tex]\( \text{Au-198} \)[/tex]) needs to be ordered so that, after decay during shipping, there will be exactly 15 mg available for use.
Here's the step-by-step breakdown:
1. Half-Life Understanding: The half-life of a substance is the time it takes for half of the substance to decay. For Au-198, the half-life is 2.69 days.
2. Decay During Shipping: The shipping time is given as 72 hours. To use the same units as the half-life, convert this time into days:
[tex]\[
72 \text{ hours} = \frac{72}{24} = 3 \text{ days}
\][/tex]
3. Decay Formula: To account for the decay over the shipping period, we use the decay formula:
[tex]\[
N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T}}
\][/tex]
where:
- [tex]\( N(t) \)[/tex] is the final amount needed (15 mg),
- [tex]\( N_0 \)[/tex] is the initial amount to be ordered,
- [tex]\( t \)[/tex] is the time period (3 days),
- [tex]\( T \)[/tex] is the half-life (2.69 days).
Rearranging the formula to find [tex]\( N_0 \)[/tex]:
[tex]\[
N_0 = \frac{N(t)}{\left(\frac{1}{2}\right)^{\frac{t}{T}}}
\][/tex]
4. Calculating the Initial Mass: Substitute the known values:
[tex]\[
N_0 = \frac{15}{\left(\frac{1}{2}\right)^{\frac{3}{2.69}}}
\][/tex]
5. Final Computation: Solve the above expression to find the initial mass that needs to be ordered. This calculation gives:
[tex]\[
N_0 \approx 32.4947 \text{ mg}
\][/tex]
Thus, the hospital should order approximately 32.495 mg of Au-198 to ensure that 15 mg is available for treatment after 3 days of shipping.
Here's the step-by-step breakdown:
1. Half-Life Understanding: The half-life of a substance is the time it takes for half of the substance to decay. For Au-198, the half-life is 2.69 days.
2. Decay During Shipping: The shipping time is given as 72 hours. To use the same units as the half-life, convert this time into days:
[tex]\[
72 \text{ hours} = \frac{72}{24} = 3 \text{ days}
\][/tex]
3. Decay Formula: To account for the decay over the shipping period, we use the decay formula:
[tex]\[
N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T}}
\][/tex]
where:
- [tex]\( N(t) \)[/tex] is the final amount needed (15 mg),
- [tex]\( N_0 \)[/tex] is the initial amount to be ordered,
- [tex]\( t \)[/tex] is the time period (3 days),
- [tex]\( T \)[/tex] is the half-life (2.69 days).
Rearranging the formula to find [tex]\( N_0 \)[/tex]:
[tex]\[
N_0 = \frac{N(t)}{\left(\frac{1}{2}\right)^{\frac{t}{T}}}
\][/tex]
4. Calculating the Initial Mass: Substitute the known values:
[tex]\[
N_0 = \frac{15}{\left(\frac{1}{2}\right)^{\frac{3}{2.69}}}
\][/tex]
5. Final Computation: Solve the above expression to find the initial mass that needs to be ordered. This calculation gives:
[tex]\[
N_0 \approx 32.4947 \text{ mg}
\][/tex]
Thus, the hospital should order approximately 32.495 mg of Au-198 to ensure that 15 mg is available for treatment after 3 days of shipping.
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