In an animal hospital, 8 units of a sulfate were injected...

In an animal hospital, 8 units of a sulfate were injected into a dog. After $50 \mathrm{~min}-$ utes, only 4 units remained in the dog. Let $f(t)$ be the amount of sulfate present after $t$ minutes. At any time, the rate of change of $f(t)$ is proportional to the value of $f(t)$. Find the formula for $f(t)$.

Key Concepts

Exponential Decay

Exponential decay is a process where the rate at which a quantity decreases is directly proportional to the current amount of that quantity. This concept is frequently modeled using the equation f(t) = f(0) e^(kt), where k is a negative constant in decay processes, representing the rate at which the substance decreases over time.

First-order Linear Differential Equations

A first-order linear differential equation is one in which the derivative of the function is proportional to the function itself. It takes the form df/dt = kf(t), and its general solution is given by f(t) = f(0) e^(kt). This framework is essential for modeling processes like exponential decay and growth.

Initial Conditions

Initial conditions provide the starting value of the function, which is used to determine the specific solution to a differential equation. In exponential models, knowing the initial quantity f(0) allows for the formulation of the exact exponential function that describes the process.

Rate Constant Determination

The rate constant, often denoted by k, quantifies the speed at which a process takes place. It can be found by substituting known values of the function at a specific time into the exponential model and solving for k, typically using logarithmic operations if the equation is in exponential form.

Solving Exponential Equations

Solving exponential equations involves working with expressions where the variable appears in an exponent. Techniques such as taking the natural logarithm of both sides are used to isolate the variable or constant in the exponent, enabling the determination of unknown parameters in the equation.

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