Question

the half-life of a radioactive kind of lead is 3 hours. if you start with 48,320 grams of it, how much will be left after 15 hours?

______ grams

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Explanation:

Step1: Find the number of half - lives

The half - life is 3 hours and the total time is 15 hours. To find the number of half - lives \(n\), we use the formula \(n=\frac{\text{total time}}{\text{half - life}}\). So \(n = \frac{15}{3}=5\).

Step2: Use the radioactive decay formula

The formula for radioactive decay is \(A = A_0\times(\frac{1}{2})^n\), where \(A_0\) is the initial amount, \(n\) is the number of half - lives, and \(A\) is the amount remaining. Here, \(A_0 = 48320\) grams and \(n = 5\). So we calculate \(A=48320\times(\frac{1}{2})^5\). First, \((\frac{1}{2})^5=\frac{1}{32}\). Then \(A = 48320\times\frac{1}{32}\). We can also calculate \(48320\div32\). \(48320\div32 = 1510\)? Wait, no, let's do it correctly. \(32\times1500 = 48000\), \(48320-48000 = 320\), \(320\div32 = 10\), so \(1500 + 10=1510\)? Wait, no, wait \(48320\div32\): \(32\times1510=32\times(1500 + 10)=32\times1500+32\times10 = 48000+320 = 48320\). Wait, but let's check the formula again. Wait, \((\frac{1}{2})^5=\frac{1}{32}\), so \(48320\times\frac{1}{32}=\frac{48320}{32}\). Let's divide numerator and denominator by 16: \(48320\div16 = 3020\), \(32\div16 = 2\), so \(\frac{3020}{2}=1510\)? Wait, no, that's wrong. Wait, \(48320\div32\): \(32\times1500 = 48000\), \(48320 - 48000=320\), \(320\div32 = 10\), so \(1500 + 10 = 1510\)? Wait, but let's do it step by step. \(48320\div2 = 24160\) (after 1 half - life), \(24160\div2 = 12080\) (after 2 half - lives), \(12080\div2 = 6040\) (after 3 half - lives), \(6040\div2 = 3020\) (after 4 half - lives), \(3020\div2 = 1510\) (after 5 half - lives). Wait, but that seems off. Wait, no, wait \(48320\div32\): \(32\times1510 = 48320\), yes. Wait, but let's check the exponent again. The number of half - lives is \(15\div3 = 5\), so the formula is \(A = 48320\times(1/2)^5=48320\times1/32 = 1510\). Wait, but maybe I made a mistake in the division. Wait, \(48320\div32\): 32*1500 = 48000, 48320 - 48000 = 320, 320/32 = 10, so 1500+10 = 1510. Yes, that's correct.

Answer:

1510