# A medical lab has 16gram of sample of radioactive isotope after 6hours it found that 12gm of sample have decayed the half life of isotope is?

General equation for radioactivity is

Where

Amount of radioactive isotope left after 6 hours is, 16 g - 12 g = 4 g

Apply natural log on both the sides

ALTERNATIVE APPROACH

Where

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The half-life of the radioactive isotope can be calculated using the formula for exponential decay:

[ N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{\frac{1}{2}}}} ]

Where:

- ( N(t) ) is the amount of the radioactive isotope remaining after time ( t ),
- ( N_0 ) is the initial amount of the radioactive isotope,
- ( T_{\frac{1}{2}} ) is the half-life of the radioactive isotope.

Given:

- ( N(t) = 12 ) grams (amount decayed after 6 hours),
- ( N_0 = 16 ) grams (initial amount),
- ( t = 6 ) hours.

Plug the given values into the formula and solve for ( T_{\frac{1}{2}} ):

[ 12 = 16 \times \left(\frac{1}{2}\right)^{\frac{6}{T_{\frac{1}{2}}}} ]

[ \frac{12}{16} = \left(\frac{1}{2}\right)^{\frac{6}{T_{\frac{1}{2}}}} ]

[ \frac{3}{4} = \left(\frac{1}{2}\right)^{\frac{6}{T_{\frac{1}{2}}}} ]

[ \frac{3}{4} = \left(\frac{1}{2}\right)^{\frac{6}{T_{\frac{1}{2}}}} = \left(2^{-1}\right)^{\frac{6}{T_{\frac{1}{2}}}} = 2^{-\frac{6}{T_{\frac{1}{2}}}} ]

[ 2^{\frac{6}{T_{\frac{1}{2}}}} = \frac{4}{3} ]

[ \frac{6}{T_{\frac{1}{2}}} = \log_2\left(\frac{4}{3}\right) ]

[ T_{\frac{1}{2}} = \frac{6}{\log_2\left(\frac{4}{3}\right)} ]

Calculate the value of ( T_{\frac{1}{2}} ) to find the half-life of the isotope.

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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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