Question

Twenty percent of a radioactive substance decays in ten years by what percent does the substance decay each year

Answer 1

The substance decays $\boxed{2.2\% }$ each year if twenty percent of a radioactive substance decays in ten year.

Further explanation:

The exponential decay formula can be expressed as follows,

$\boxed{y = a{{\left( {1 - r} \right)}^x}}$

Here, $a$ represents the initial amount, $y$ is the final amount,$r$ is the rate of decay, and $x$ represents the time.

Given:

Twenty percent of a radioactive substance decays in ten year.

Explanation:

Consider the initial amount of the radioactive substance be $A.$

The final amount of the radioactive substance is $0.80A$

The rate of decay can be obtained as follows,

$\begin{aligned}0.80A &= A{\left( {1 - r} \right)^{10}}\\\frac{{0.80A}}{A} &= {\left( {1 - r} \right)^{10}}\\0.80 &= {\left( {1 - r} \right)^{10}}\\\sqrt[{10}]{{0.80}} &= 1 - r\\r&= 1 - \sqrt[{10}]{{0.80}}\\r&= 1 - 0.978\\r&= 0.022\\\end{aligned}$

The rate of decay is $2.2\%.$

The substance decays $\boxed{2.2\% }$ each year if twenty percent of a radioactive substance decays in ten year.

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Exponential decay

Keywords: twenty percent, radioactive substance, decays, ten years, each year, rate of decay each year, substance.

Answer 2

If the substance is decaying at such a rate, it will be decaying at a rate of 2% per year.

The radioactive substance decays at 20% every 10 years.

In order to find the rate it decays in a single year, use:

= Rate of decay / Number of years

= 20% / 10

= 2%

In conclusion, the substance decays at 2% per year.

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