Problem 76 - Algebra and Trigonometry

Question

Skydiving The velocity of a sky diver \(t\) seconds after jumping is given by \(v(t)=80\left(1-e^{-0.2 t}\right) .\) After how many seconds is the velocity 70 \(\mathrm{ft} / \mathrm{s}\) ?

Step-by-Step Solution

Answer

The velocity reaches 70 ft/s after approximately 10.4 seconds.

Step 1: Understand the Given Function

The velocity function provided is \( v(t) = 80(1 - e^{-0.2t}) \). This describes the velocity of a sky diver as a function of time \( t \). Our task is to find the value of \( t \) when the velocity \( v(t) = 70 \text{ ft/s} \).

Step 2: Set Up the Equation

Start by setting the velocity function equal to 70 ft/s: \( 80(1 - e^{-0.2t}) = 70 \). This equation will allow us to solve for \( t \).

Step 3: Isolate the Exponential Term

Divide both sides of the equation by 80 to isolate the exponential term: \( 1 - e^{-0.2t} = \frac{70}{80} \). Simplify the fraction to get \( 1 - e^{-0.2t} = 0.875 \).

Step 4: Solve for the Exponential Term

Subtract 1 from both sides to isolate \( e^{-0.2t} \): \( -e^{-0.2t} = 0.875 - 1 \). Simplifying gives \( -e^{-0.2t} = -0.125 \). Multiply both sides by -1 to obtain \( e^{-0.2t} = 0.125 \).

Step 5: Take the Natural Logarithm

Use the natural logarithm to solve for \( t \): take the natural log on both sides: \( \ln(e^{-0.2t}) = \ln(0.125) \). This simplifies to \( -0.2t = \ln(0.125) \).

Step 6: Solve for t

Divide both sides by -0.2 to solve for \( t \): \( t = \frac{\ln(0.125)}{-0.2} \). Calculating the natural logarithm gives \( \ln(0.125) \approx -2.079 \). Thus, \( t \approx \frac{-2.079}{-0.2} = 10.395 \).

Step 7: Round to Appropriate Decimal

Since the velocity will be measured in seconds, it is appropriate to round \( t \) to a reasonable decimal place: \( t \approx 10.4 \) seconds.

Key Concepts

Velocity Function

In the realm of physics and motion, a velocity function describes how the velocity of an object changes over time. Specifically, in the case of the skydiver, the velocity function is given by \( v(t) = 80(1 - e^{-0.2t}) \). Here's what each component means:

• The constant 80 represents the terminal velocity the skydiver approaches. This is essentially the maximum speed the skydiver will reach under constant conditions, such as gravity and drag.
• The expression \(1 - e^{-0.2t}\) encapsulates the time-dependent change in velocity, governed by an exponential decay component, \( e^{-0.2t} \).
• As time \(t\) increases, the term \(-e^{-0.2t}\) tends toward zero, making the entire factor \(1 - e^{-0.2t}\) approach 1, hence bringing the velocity \(v(t) \) closer to 80 ft/s.

With this understanding, we interpret the velocity function graphically as a curve that steeply ascends and gradually levels off as time progresses. It models how quickly the skydiver reaches terminal velocity after jumping.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is an essential mathematical function when solving exponential equations. It is the inverse of the exponential function involving Euler's number \( e \), which is approximately equal to 2.71828. To solve for time \( t \) when the skydiver's velocity is set to a specific value, we utilize the properties of the natural logarithm.
When dealing with equations like \( e^{-0.2t} = 0.125 \), the natural logarithm allows us to strip away the exponential function. By applying \( \ln \) to both sides, it becomes \( \ln(e^{-0.2t}) = \ln(0.125) \).
Thanks to the property \( \ln(e^x) = x \), the left side simplifies to \( -0.2t \), thus yielding \( -0.2t = \ln(0.125) \). This simplification is powerful as it converts an otherwise complex exponential situation into a linear equation that we can easily solve for \( t \).

Problem Solving

Solving for time in the velocity function challenge involves a clear problem-solving sequence. Simply put, we start with understanding the problem, setting equations, and logically working through to our solution. Here's a breakdown of the steps:

• First, clearly identify the provided variable values and unknowns. For this exercise, the given velocity is 70 ft/s and time \( t \) is unknown.
• Secondly, form an equation by substituting known values into the velocity function: \( 80(1 - e^{-0.2t}) = 70 \).
• Rearrange and simplify: Dividing each side by 80, and isolating the term \( e^{-0.2t} = 0.125 \).
• Apply the natural logarithm to solve for \( t \): use \( \ln(e^{-0.2t}) = \ln(0.125) \), simplifying to \( -0.2t = \ln(0.125) \).
• Calculate \( t \): divide both sides by -0.2 leading to \( t = \frac{\ln(0.125)}{-0.2} \), resulting in approximately 10.4 seconds.

This structured approach of breaking down the problem ensures clarity and allows us to reach the solution efficiently. Remember always to check your intermediate results at each stage to avoid errors in the final result.