Question

The fraction of a radioactive material which remains active after time $t$ is $9 / 16 .$ The fraction which remains active after time $t / 2$ will be
A $\dfrac{4}{5}$
B. $\dfrac{7}{8}$
C. $\dfrac{3}{5}$
D $\dfrac{3}{4}$

Answer 1

Hint: We know that a time period (denoted by 'T’) is the time taken for one complete cycle of vibration to pass a given point. As the frequency of a wave increases, the time period of the wave decreases. The unit for time period is 'seconds'. We can say that frequency is the number of occurrences of a repeating event per unit of time. It is also referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency. Frequency is measured in units of hertz (Hz) which is equal to one occurrence of a repeating event per second. Frequency measures the number of times something occurs in a specific amount of time. While frequency can be used to measure the rate of any action, in technical applications it is typically used to measure wave rates or processing speed.

Complete step by step answer
We can say that radioactivity refers to the particles which are emitted from nuclei as a result of nuclear instability. Because the nucleus experiences the intense conflict between the two strongest forces in nature, it should not be surprising that there are many nuclear isotopes which are unstable and emit some kind of radiation.
First order decay:
$\mathrm{N}(\mathrm{t})=\mathrm{N}_{\mathrm{o}} \mathrm{e}^{-\lambda \mathrm{t}}$
Given: $\mathrm{N}(\mathrm{t}) / \mathrm{N}_{\mathrm{o}}=9 / 16$
Now, $\mathrm{N}(\mathrm{t} / 2)=\mathrm{N}_{\mathrm{o}} \mathrm{e}^{-\lambda \mathrm{t} / 2}$
$\dfrac{\mathrm{N}(\mathrm{t} / 2)}{\mathrm{N}_{\mathrm{o}}}=\sqrt{\mathrm{e}^{-\lambda \mathrm{t}}}=\sqrt{9 / 16}$
$\mathrm{N}(\mathrm{t} / 2)=\dfrac{3}{4} \mathrm{N}_{\mathrm{o}}$

So the correct option is option D.

Note: We know that wave frequency is the number of waves that pass a fixed point in a given amount of time. The SI unit for wave frequency is the hertz (Hz), where 1 hertz equals 1 wave passing a fixed point in 1 second. T is the time it takes for one complete oscillation; it is measured in seconds. All waves, including sound waves and electromagnetic waves, follow this equation. For example, a wave with a time period of 2 seconds has a frequency of 1/ 2 = 0.5 Hz. A sound wave has a time period of 0.0001 seconds.