Question

Element X decays radioactively with a half life of 13 minutes. If there are 430 grams of Element X, how long, to the nearest tenth of a minute, would it take the element to decay to 6 grams?

Answer 1

Using an exponential function, it is found that it would take 80.12 minutes for the element to decay to 6 grams.

What is an exponential function?

An exponential function is defined according to the following rule:

$y = a(b)^{\frac{t}{n}}$

In which:

• a is the initial value of the function.

• b is the rate of change of the function.

• n is the time in which the rate of change occurs.

Considering the half life of the substance, and the initial amount of 430 grams, the parameters are given as follows:

a = 430, b = 0.5, n = 13.

Then the amount of the substance after t minutes is given by:

$y = 430(0.5)^{\frac{t}{13}}$

The time it will take for the element to 6 grams is t when y = 6, hence:

$6 = 430(0.5)^{\frac{t}{13}}$

$(0.5)^{\frac{t}{13}} = \frac{6}{430}$

$\log{((0.5)^{\frac{t}{13}})} = \log{\left(\frac{6}{430\right)}$

$\frac{t}{13}\log{0.5} = \log{\left(\frac{6}{430\right)}$

$t = 13\frac{\log{\left(\frac{6}{430}\right)}}{\log{0.5}}$

t = 80.12 minutes.

More can be learned about exponential functions at brainly.com/question/25537936

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