Problem 40 - Precalculus: Mathematics for Calculus

Question

The downward velocity of a falling raindrop at time \(t\) is modeled by the function $$v(t)=1.2\left(1-e^{-8.2 t}\right)$$ (a) Find the terminal velocity of the raindrop by evaluating \(\lim _{t \rightarrow \infty} v(t) .\) (Use the result of Example 3.) (b) Graph \(v(t),\) and use the graph to estimate how long it takes for the velocity of the raindrop to reach \(99 \%\) of its terminal velocity.

Step-by-Step Solution

Answer

(a) Terminal velocity is 1.2. (b) Velocity reaches 99% of terminal value in about a few seconds.

Step 1: Evaluate the Limit for Terminal Velocity

To find the terminal velocity of the raindrop, we need to evaluate the limit of \(v(t)\) as \(t\) approaches infinity. The function given is \(v(t) = 1.2(1 - e^{-8.2t})\). Consider \(\lim_{t \rightarrow \infty} e^{-8.2t}\), since the exponential term \(e^{-8.2t}\) approaches 0 as \(t\) increases, the overall expression becomes:\[ \lim_{t \rightarrow \infty} v(t) = 1.2(1 - 0) = 1.2 \].

Step 2: Graph the Downward Velocity Function

Sketch the graph of \(v(t) = 1.2(1 - e^{-8.2t})\). The graph is an exponentially increasing curve that levels off as it approaches the terminal velocity \(1.2\). The key feature to highlight in the graph is its asymptotic behavior as \(t\) becomes very large.

Step 3: Calculate 99% of Terminal Velocity

99% of the terminal velocity can be found by multiplying the terminal velocity by 0.99. Thus, \(0.99 \times 1.2 = 1.188\). This is the target velocity we need to estimate from the graph.

Step 4: Use Graph to Estimate Time

By referring to the graph of \(v(t)\), observe the point where the velocity reaches approximately \(1.188\). This involves visually estimating the time value \(t\) where the function \(1.2(1 - e^{-8.2t})\) first reaches \(1.188\). Typically, this might occur after just a few seconds given the rate of the exponential decline.

Key Concepts

Exponential Functions

In understanding the velocity of a raindrop as it falls, the function involved is an exponential function. It's written as \(v(t) = 1.2(1 - e^{-8.2t})\). This type of function is commonly used in situations where something changes rapidly at first and then slows down, like the speed of a falling raindrop approaching its terminal velocity.
Exponential functions have a base number, here \(e\), which is the natural exponential base approximately equal to 2.718. The exponent is a product of a negative constant and the variable \(t\), which is time in this context. It describes how fast the exponent causes the change. The rate \(-8.2t\) for \(t\) indicates a rapid decline towards a fixed value, visibly seen in the decay term \(e^{-8.2t}\) going to zero as \(t\) increases. Thus, the velocity function approaches the constant multiplier, 1.2, which signifies the terminal velocity.
When working with exponential functions, always remember to interpret:

• The constant before the base \(e\): impacts the final steady-state value.
• The sign and size of the exponent: shows increase or decline speed.

Limits in Calculus

To determine the terminal velocity of the raindrop, we use the concept of limits in calculus. A limit assesses what value a function approaches as the input goes to some point, often infinity in these scenarios. For the raindrop's velocity \(v(t) = 1.2(1 - e^{-8.2t})\), we analyze \(\lim_{t \to \infty} v(t)\).
The key here is understanding how limits comprehend the behavior of functions on long-term scales. As \(t\) approaches infinity, \(e^{-8.2t}\) tends towards zero because an exponentially decreasing term in the format \(e^{\text{negative value}}\) reduces rapidly. Thus, the limit essentially simplifies to:

• Removing the diminishing exponential part when \(t\) is large.
• A simple algebraic expression of the constant, indicating the result.

For our function, this analysis shows the velocity stabilizes at 1.2. Limits help predict steady states or asymptotic behaviors essential in calculus and dynamics.

Graphing Functions

Graphing functions helps visually interpret behaviors like how velocity changes over time. Consider the graph of \(v(t) = 1.2(1 - e^{-8.2t})\), a type illustrating exponential rise reaching a horizontal line, or asymptote, reflecting the terminal velocity.
Here’s a guide to understanding and plotting this graph:

• Begin by identifying the horizontal asymptote at \(1.2\), indicating the terminal velocity the graph approaches.
• Notice the graph's initial steep rise, suggesting quick changes due to the exponential factor \(e^{-8.2t}\).
• The curve flattens and closely parallels the asymptote as \(t\) grows, displaying slowing changes.

By graphing, students can estimate visually aspects like how long it takes for a raindrop's velocity to reach 99% of its terminal velocity. Use this graphical representation to interpret complex phenomena simply and effectively. Locate points of interests such as when the velocity nears \(0.99 \times 1.2 = 1.188\), which translates to time on the horizontal axis, making estimations such as this more intuitive.