Assignment #5: Exploring Calculus Applications
Task 1: Choose a Topic
Topic Selected: Related Rates
Problem Scenario:
A hot air balloon is rising vertically from the ground at a constant rate of 3 meters per minute. A person is standing at a point on the ground that is 400 meters away from the point directly underneath the balloon. We need to determine how fast the distance between the person and the balloon is changing when the balloon is 100 meters above the ground.
Rates Explored:
- The height \( h \) of the balloon is increasing at a rate of \( \frac{dh}{dt} = 3 , \text{m/min} \).
- The horizontal distance \( x \) from the person to the balloon remains constant at \( 400 , \text{m} \).
- The distance \( D \) between the person and the balloon is what we need to find, particularly the rate of change \( \frac{dD}{dt} \).
Description of Rates:
- The height of the balloon \( h \) is changing with time as it rises.
- The horizontal distance \( x \) is fixed.
- We are interested in how quickly the distance \( D \) changes as a function of the balloon's height.
Solution Steps:
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Identify relationships: The relationship is given by the Pythagorean theorem: \[ D^2 = x^2 + h^2 \] where:
- \( D \) is the distance from the person to the balloon.
- \( x = 400 , \text{m} \)
- \( h = \text{height of the balloon (in m)} \)
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Differentiate with respect to time: Differentiate both sides of the equation with respect to time \( t \): \[ 2D \frac{dD}{dt} = 0 + 2h \frac{dh}{dt} \]
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Solve for \( \frac{dD}{dt} \): Rearrange to find: \[ \frac{dD}{dt} = \frac{h}{D} \frac{dh}{dt} \]
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Determine specific values: When the balloon is \( h = 100 , \text{m} \) and \( x = 400 , \text{m} \): First calculate \( D \): \[ D = \sqrt{x^2 + h^2} = \sqrt{400^2 + 100^2} = \sqrt{160000 + 10000} = \sqrt{170000} \approx 412.31 , \text{m} \] Now substitute the values into the rate equation: \[ \frac{dD}{dt} = \frac{100}{412.31} (3) \approx \frac{300}{412.31} \approx 0.727 , \text{m/min} \]
Conclusion:
Thus, when the hot air balloon is 100 meters above the ground, the distance between the person and the balloon is increasing at a rate of approximately \( 0.727 , \text{m/min} \).
Task 2: Present Your Activity
To prepare for your submission, you can create a PDF document or slide deck that includes:
- The full problem statement and solution, formatted with proper mathematical notation.
- Diagrams illustrating the scenario, including a right triangle showing the balloon's height, the horizontal distance, and the resultant distance.
Example Diagram:
Balloon Height (h)
|
| (D)
|\
| \
| \
| \
| \
| \
| \
|_______\
400 m
Task 3: Cite Your Sources ([5 marks])
- Thomas, G. B., & Finney, R. L. (2006). Calculus and Analytic Geometry (9th ed.). Pearson Education.
- Stewart, J. (2015). Calculus: Early Transcendentals (7th ed.). Cengage Learning.
- Hughes, R. (2018). Calculus: Graphical, Numerical, Algebraic (5th ed.). Key Curriculum Press.
Make sure to follow APA format for your citations, including proper formatting for books and articles.
This completes your assignment outline for exploring the application of calculus in the context of related rates. You can expand your work by applying similar structures for tasks related to optimization and vectors as required.