Solve Related Rates Problems: Problem 1 & 2

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SUMMARY

This discussion focuses on solving two advanced related rates problems involving geometric figures: a circle inscribed in a square and a tightrope walker casting a shadow. In Problem 1, the rate at which the perimeter of the square increases is derived from the circle's circumference increasing at 6 inches per second, while the area between the circle and square is analyzed when the circle's area is 25(pi) square inches. Problem 2 examines the dynamics of a tightrope walker moving at 2 feet per second and the movement of her shadow along the ground and up the wall of a building.

PREREQUISITES
  • Understanding of related rates in calculus
  • Familiarity with geometric properties of circles and squares
  • Knowledge of basic trigonometry for shadow problems
  • Ability to differentiate functions with respect to time
NEXT STEPS
  • Study the concept of related rates in calculus with examples
  • Learn how to derive relationships between geometric figures
  • Explore applications of related rates in real-world scenarios
  • Practice solving problems involving shadows and angles in trigonometry
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Students studying calculus, particularly those focusing on related rates, as well as educators looking for problem-solving strategies in geometry and trigonometry.

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Hello. I really need help on my math homework. Please anybody help me. I would really appreciate it. These 2 problems have to do with related rates and they are too advanced for me... I just need to be shown the way and I could get it I just need to know how to start... thank you.

Problem 1
A circle is inscribed in a square. The circumference of the circle is increasing at a constant rate of 6 inches per second. As the circle expands, the square expands to maintain the condition of tangency. (Note: A circle with radius r has circumference C = 2(pi)r and Area A = (pi) r^2.)
a) Find the rate at which the perimeter of the square is increasing. Indicate units of measure.
b) At the instant when the area of the circle is 25(pi) square inches, find the rate of increasing in the area enclosed between the circle and the square. Indicate units of measurement.

Problem 2
A tightrope is stretched 30 feet above the ground between the Jay (J) and the Tee (T) buildings, which are 50 feet apart. A tightrope walker, walking at a constant rate of 2 feet per second from point A to point B, is illuminated by a spotlight 70 feet above point A, as shown in the diagram.

a) How fast is the shadow of the tightrope walkers feet moving along the ground when she is midway between the buildings?
b) How far from point A is the tightrope walker when the shadow of her feet reaches the base of the Tee Building?
c) How fast is the shadow of the tightrope walker's feet moving up the wall of the Tee building when she is 10 feet from point B?
 
Physics news on Phys.org
1.a. What is the relation of the square's circumference to the circle's?
1.b. What is the relation of the square's area to the circle's?
 

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