# The half-life of strontium-90 is 28 years. How long will it take a 44 mg sample to decay to a mass of 11 mg?

The nuclear half-life of a radioactive isotope indicates how long it will take for an initial sample to be halved.

and so forth.

Take note that you can use the number of half-lives that pass to write the remaining portion of an initial sample.

This indicates that the decay of the strontium-90 sample to a quarter of its initial mass requires the passage of two half-lives.

This suggests that you have

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To calculate the time it takes for a sample of strontium-90 to decay to a certain mass, you can use the formula:

[ N(t) = N_0 \cdot \left(\frac{1}{2}\right)^\frac{t}{t_{1/2}} ]

Where:

- ( N(t) ) is the final mass of the sample
- ( N_0 ) is the initial mass of the sample
- ( t ) is the time elapsed
- ( t_{1/2} ) is the half-life of strontium-90

Given:

- ( N_0 = 44 ) mg
- ( N(t) = 11 ) mg
- ( t_{1/2} = 28 ) years

Solving for ( t ):

[ 11 = 44 \cdot \left(\frac{1}{2}\right)^\frac{t}{28} ]

[ \left(\frac{1}{4}\right) = \left(\frac{1}{2}\right)^\frac{t}{28} ]

[ \log_2\left(\frac{1}{4}\right) = \frac{t}{28} ]

[ -2 = \frac{t}{28} ]

[ t = -2 \times 28 ]

[ t = -56 ]

However, time cannot be negative. Therefore, the equation indicates that 56 years have passed. Therefore, it takes 56 years for the 44 mg sample to decay to a mass of 11 mg.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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