A laboratory sample contains 1.6 \times 10^{17} atoms of a radioactive isotope. The sample… — ACT

A laboratory sample contains $1.6 \times 10^{17}$ atoms of a radioactive isotope. The sample decays such that at the end of each 6-hour interval, the number of remaining atoms is equal to the square root of the number of atoms that were present at the beginning of that interval. Which of the following is the number of atoms remaining in the sample at the end of 12 hours?

A
$4 \times 10^4$
B
$2 \times 10^{4.25}$
$2 \times 10^4$ Answer
D
$4 \times 10^8$
E
$4 \times 10^{16}$

Answer

2 \times 10^4

The correct option is

2 \times 10^4

. To find the number of remaining atoms after 12 hours, we must apply the square root operation twice, since 12 hours consists of two 6-hour intervals. First, rewrite the initial number of atoms as

16 \times 10^{16}

to make the exponent even. After 6 hours, the number of atoms is

\sqrt{16 \times 10^{16}} = 4 \times 10^8

. After another 6 hours, we take the square root again:

\sqrt{4 \times 10^8} = 2 \times 10^4

Step-by-Step Solution

Determine the number of 6-hour intervals in a 12-hour period.

There are

12 \div 6 = 2

intervals.

Since the population is reduced to its square root every 6 hours, we must apply the square root operation twice over a 12-hour period.

Rewrite the initial number of atoms in a form that simplifies taking the square root.

1.6 \times 10^{17} = 16 \times 10^{16}

Converting the coefficient to a perfect square and the exponent to an even number allows us to easily compute the square root without a calculator.

Calculate the number of atoms remaining after the first 6-hour interval by taking the square root of the initial value.

\sqrt{16 \times 10^{16}} = \sqrt{16} \times \sqrt{10^{16}} = 4 \times 10^8

The square root of a product is the product of the square roots, and the square root of

10^{16}

10^{16 \cdot 0.5} = 10^8

Calculate the number of atoms remaining after the second 6-hour interval (12 hours total) by taking the square root of the value at the end of the first interval.

\sqrt{4 \times 10^8} = \sqrt{4} \times \sqrt{10^8} = 2 \times 10^4

Applying the square root operation to the intermediate quantity of

4 \times 10^8

yields the final remaining atoms after the full 12 hours.

Key Concept

Applying square root operations to expressions written in scientific notation.

Alternative Method

We can compute the overall decay multiplier first. Taking the square root twice is equivalent to raising the initial quantity to the power of

\frac{1}{4}

. Thus, the final quantity is

(1.6 \times 10^{17})^{1/4} = (16 \times 10^{16})^{1/4} = 16^{1/4} \times (10^{16})^{1/4} = 2 \times 10^4

Estimated Time:2m 0s

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