How Do You Solve Sphere Inscription and Shadow Length Problems?

In summary, the conversation discusses two problems - calculating the altitude of an inscribed right circular cylinder within a sphere and determining the rate of change of a man's shadow as he walks away from a light source. The first problem can be simplified by focusing on the first quadrant of a circle and maximizing the area of a rectangle inscribed within it. The second problem involves using Pythagoras and similar triangles.
  • #1
Kobrakai
2
0
I am having trouble with these two problems, I was wondering if anyone here could help me.

1. Given a sphere of radius 10 inches. Calculate the altitude of the inscribed right circular cylinder of maximum volume.

2. A man 6 feet tall walks away from a light 30 feet high at the rate of 3 miles per hour. How fast is the further end of his shadow moving, and how fast is his shadow lengthening?
 
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  • #2
this really belongs in the calculus forum.

for the first problem, you can simplify it by looking at only the section of a circle in the first quadrant of the coordinate plane. then look to maximize the area of a rectangle inscribed within the area of the 1/4 circle.

the second problem reminds me that I haven't done a related rates problem in some 6 months, and don't feel like brushing up at the moment :-)
 
  • #3
these are pythagoras and similar triangles problems.
 

1. What are sphere and shadow problems?

Sphere and shadow problems involve understanding the relationship between a sphere and its shadow when it is illuminated by a light source. These types of problems often involve geometry, trigonometry, and concepts from optics.

2. Why are sphere and shadow problems important?

Understanding how a sphere and its shadow interact is important in various fields such as astronomy, architecture, and art. It can also help in practical applications such as designing solar panels or predicting the movement of celestial objects.

3. What are some common techniques for solving sphere and shadow problems?

Some common techniques for solving sphere and shadow problems include using geometric formulas, applying trigonometric functions, and using concepts from optics such as the law of reflection. It is also helpful to draw diagrams and use algebra to solve equations.

4. What are some challenges in solving sphere and shadow problems?

One of the main challenges in solving sphere and shadow problems is visualizing the 3D relationship between the sphere, light source, and shadow. It can also be challenging to apply the correct formulas and concepts to solve the problem accurately.

5. Are there any practical applications of solving sphere and shadow problems?

Yes, there are many practical applications of solving sphere and shadow problems. As mentioned earlier, it can be used in fields such as astronomy, architecture, and art. It can also be applied in engineering, physics, and computer graphics to create realistic simulations and models.

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