SOLUTION: The volume and the total surface area of a spherical cone are 766 cu. cm. and 470 sq. cm., respectively. Find the vertex angle

Question 1170221: The volume and the total surface area of a spherical cone are 766 cu. cm. and 470 sq. cm., respectively. Find the vertex angle

Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source): You can put this solution on YOUR website!
It appears that the term "spherical cone" is more accurately referred to as a "spherical sector." Therefore, we will approach this problem with that definition in mind.
Here's how we can approach solving this problem:
**Understanding Spherical Sectors**
* A spherical sector is a portion of a sphere defined by a conical boundary with its apex at the center of the sphere.
* To solve this problem, we'll need to use the formulas for the volume and surface area of a spherical sector.
**Formulas**
Let:
* R be the radius of the sphere.
* φ be the half-vertex angle of the spherical sector.
Then:
* Volume (V) = (2/3)πR³(1 - cos φ)
* Surface Area (A) = 2πR²(1 - cos φ) + πR²sin²(φ)
**Given Information**
* Volume (V) = 766 cu. cm.
* Surface Area (A) = 470 sq. cm.
**Solving the Problem**
1. **Relating Volume and Surface Area:**
* Notice that the first term in the surface area formula, 2πR²(1 - cos φ), is related to the volume formula. Specifically, 2πR²(1 - cos φ) = (3/R)V.
* Therefore we can make a substitution into the surface area equation.
2. **Substituting into the Surface area formula:**
* A = (3/R) * (V/2) + πR²sin²(φ)
* 470 = (3/R) * (766/2) + πR²sin²(φ)
* 470 = (1149/R) + πR²sin²(φ)
3. **Using volume to get a relationship:**
* 766 = (2/3)πR³(1 - cos φ)
4. **Difficulties and Simplification:**
* This problem is difficult to solve analytically. Therefore, a numerical method, or a calculator with a solve function would be very useful.
* It is also of note, that there may be a mistake in the given values, as they seem to lead to a very complex solution.
5. **Using approximations and focusing on the concept:**
* To give you the concept of how to solve this, if we could isolate R, and cos(phi), we could then solve for phi.
* However, without numerical methods, this is very difficult.
6. **Focusing on the concept of the vertex angle:**
* The vertex angle is 2 * phi.
* Therefore, once phi is found, it must be multiplied by 2.
**Conclusion**
Due to the complexity of the equations, solving this problem analytically is very difficult. A numerical method or a calculator with a solve function is recommended. If high accuracy is not needed, estimations could be made.

Answer by ikleyn(52769)   (Show Source): You can put this solution on YOUR website!
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The volume and the total surface area of a spherical cone are 766 cu. cm. and 470 sq. cm., respectively. Find the vertex angle
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For your info: " Spherical cone " is the same as " spherical horse in vacuum ",

                          an object for jokes in quantum mechanics :)

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