Given that $d\left(t\right)$ is a known function and $d$ is a specific value, which of the following procedures will find the time $t$ at which $d\left(t\right)=d$?

• A. Find the limit of $d\left(t\right)$ as $t$ approaches $\infty$.
• B. Integrate $d\left(t\right)$ from $0$ to $t$ and set the result equal to $d$.
• C. Differentiate $d\left(t\right)$ and set the derivative equal to $d$.
• D. Solve the equation $d\left(t\right)=d$ for $t$.

Answer: (D)

Verified step by step guidance

1. Understand the problem: We are tasked with finding the time t at which the function d(t) equals a specific value d. This means we need to solve the equation d(t) = d for t.
2. Step 1: Recognize that solving d(t) = d involves finding the value(s) of t where the function d(t) intersects the horizontal line y = d.
3. Step 2: Rearrange the equation d(t) = d, if necessary, to isolate t. This may involve algebraic manipulation or applying inverse functions, depending on the form of d(t).
4. Step 3: Check the other options provided in the problem. For example, finding the limit of d(t) as t approaches infinity, integrating d(t), or differentiating d(t) are not relevant to solving for t in this context. These operations serve different purposes in calculus.
5. Step 4: Once the equation d(t) = d is solved for t, verify the solution by substituting the value(s) of t back into the original equation to ensure that d(t) equals d.