The choice of isotope for each application is dictated by a careful consideration of the type of radiation it emits (alpha, beta, or gamma) and its half-life. The radiation type determines its ability to interact with the target material, while the half-life dictates the duration of the radioactive decay and therefore the practicality of the application.

Household Fire (Smoke) Alarms: Americium-241 (²⁴¹Am) is commonly used. It emits alpha particles. Alpha particles have a short range and are easily stopped by materials like paper or air. This is ideal because the smoke detector needs to be shielded to prevent the alpha particles from reaching the detector, but the alpha particles need to be able to penetrate the detector material to trigger the alarm. The half-life of ²⁴¹Am is 432 years, which is long enough to provide a reliable source of alpha particles for a practical lifespan of the alarm. Beta and gamma radiation would be too easily stopped or would require excessive shielding, making the alarm impractical.

Irradiating Food to Kill Bacteria: Cobalt-60 (⁶⁰Co) and Cesium-137 (¹³⁷Cs) are commonly used. Both emit gamma rays. Gamma rays have high penetrating power and can effectively kill bacteria and other microorganisms within food. The half-lives of ⁶⁰Co (5.27 years) and ¹³⁷Cs (30 years) are suitable for food irradiation processes, ensuring a sufficient level of radioactivity for effective sterilization. Alpha and beta radiation would not penetrate the food sufficiently to be effective.

Sterilisation of Equipment using Gamma Rays: Cobalt-60 (⁶⁰Co) and Cesium-137 (¹³⁷Cs) are again commonly used. Gamma rays are ideal for sterilizing medical equipment because of their high penetrating power. They can pass through packaging and materials to reach all parts of the equipment, killing bacteria, viruses, and other pathogens. The half-lives of ⁶⁰Co and ¹³⁷Cs are suitable for long-term storage and use in sterilization procedures. Alpha and beta radiation would not penetrate the equipment effectively.

Measuring and Controlling Thicknesses of Materials: Iodine-131 (¹³¹I) is often used. ¹³¹I emits gamma rays. The intensity of the gamma rays that pass through a material is related to the thickness of the material. By measuring the intensity of the transmitted gamma rays, the thickness can be accurately determined. Gamma rays have good penetration, allowing for measurements of relatively thick materials. The half-life of ¹³¹I (8.02 days) is short enough that the radiation dose is relatively low and the material doesn't become excessively radioactive over time. Alpha and beta radiation would not be suitable because they have low penetration power.

Diagnosis and Treatment of Cancer using Gamma Rays: Cobalt-60 (⁶⁰Co) and Cesium-137 (¹³⁷Cs) are used in radiotherapy. Gamma rays are used to kill cancer cells by damaging their DNA. The high penetrating power of gamma rays allows them to reach tumors located deep within the body. The half-lives of ⁶⁰Co and ¹³⁷Cs are suitable for providing a sustained source of gamma radiation for cancer treatment. Alpha and beta radiation are less effective for treating deep-seated tumors due to their limited penetration.

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Solution:

1. We know that A(t) = A₀e^-kt. We are given A₀ = 1000 and we can use the data from the decay curve to find k. Using t = 100 s and A(100) = 315.5: 315.5 = 1000e^-100k
2. Divide both sides by 1000: 0.3155 = e^-100k
3. Take the natural logarithm of both sides: ln(0.3155) = -100k => -1.152 = -100k
4. Solve for k: k = -1.152 / -100 = 0.01152 s^-1
5. The half-life (T_1/2) is given by: T_1/2 = ln(2) / k
6. Substitute the value of k: T_1/2 = 0.693 / 0.01152 = 60.1 s
7. Therefore, the half-life of the isotope is approximately 60.1 seconds.

Answer: The table above shows the number of nuclei remaining after successive 20-minute periods. Each 20-minute period represents one half-life.