Question

Certain metals are used to sterilize some medical supplies such as syringes and surgical gloves. Suppose that in a laboratory, an original 9-milligram sample of a metal decayed to 8 milligrams in 1 year. Find the half-life in years of the metalCertain metals are used to sterilize some medical supplies such as syringes and surgical gloves. Suppose that in a laboratory, an original 9-milligram sample of a metal decayed to 8 milligrams in 1 year. Find the half-life in years

Answer 1

The half-life of the metal is 18 years.

Step-by-step explanation:

The half-life of a radioactive substance is the time it takes for half of the sample to decay. In this case, the original 9-milligram sample decayed to 8 milligrams in 1 year. This means that half of the sample decayed in 1 year. To find the half-life, we can set up a proportion:

1 year is to 0.5 (half of the original sample) as the half-life is to 9 milligrams.

Cross multiplying, we get:

1 x 9 = 0.5 x half-life

9 = 0.5 x half-life

Dividing both sides by 0.5:

half-life = 9 / 0.5 = 18 years

Therefore, the half-life of the metal is 18 years.

Answer 2

The half-life of the metal, based on this decay constant, is 5.88 years.

How do we find the half-life in years?

P₀ = 9 milligrams (initial amount),

P = 8 milligrams (amount after 1 year),

t = 1 year

k is the decay constant.

P =
`P_0e^(-kt)`

8 =
`9e^(-k(1))`

8/9 =
`e^(-k)`

k = −ln(8/9)

​k = 0.11778

The half-life T of the substance can be found by the formula:

T = ln(2)/k

T= ln(2)/0.117783

T = 0.693147/0.117783

T = 5.88

​The decay constant k for the metal is approximately 0.118 per year. The half-life of the metal, based on this decay constant, is approximately 5.88 years. ​​

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