Problem 8 - Calculus Early Transcendentals

Question

Give two examples of processes that are modeled by exponential decay.

Step-by-Step Solution

Answer

Question: Provide two examples of processes that are modeled by exponential decay. Answer: Two examples of processes modeled by exponential decay are radioactive decay and the charging/discharging of a capacitor. Radioactive decay occurs when unstable atomic nuclei lose energy by emitting radiation. The number of radioactive atoms remaining after a certain time can be described by the exponential decay function N(t) = N₀e^(-λt). The charging/discharging of a capacitor in an electrical circuit involves the change in voltage across its terminals. The voltage across a discharging capacitor after a certain time can be described by the exponential decay function V(t) = V₀e^(-t/RC).

Step 1: Understand radioactive decay

Radioactive decay is a process in which unstable atomic nuclei lose energy by emitting radiation. As an unstable nucleus decays, the number of radioactive atoms decreases over time. The rate at which these atoms decay is proportional to the current quantity of radioactive atoms, making this an exponential decay process.

Step 2: Exponential decay function for radioactive decay

The number of radioactive atoms remaining after a certain time for a sample can be described by the following exponential decay function: N(t) = N₀e^(-λt), where N(t) is the number of radioactive atoms at time t, N₀ is the initial number of radioactive atoms, λ is the decay constant, and t is the time that has passed. Example 2: Charging or discharging of a capacitor

Step 1: Understand the capacitor charging/discharging process

In an electrical circuit, a capacitor stores electrical charge. When a capacitor charges, the voltage across its terminals increases, whereas during discharging, the voltage decreases. The rate at which voltage changes during charging/discharging is proportional to the difference between the current voltage and the maximum/minimum voltage, making this an exponential decay process.

Step 2: Exponential decay function for capacitor discharging

The voltage across a discharging capacitor after a certain time can be described by the following exponential decay function: V(t) = V₀e^(-t/RC), where V(t) is the voltage at time t, V₀ is the initial voltage, R is the resistance in the circuit, C is the capacitance of the capacitor, and t is the time that has passed.