Find the dimensions of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius 70 cm. What is the maximum​ volume?
Question content area bottom
Part 1
The radius is 
  
enter your response here ​cm, and the height is 
  
enter your response here cm.
​(Type exact​ answers, using radicals as​ needed.)

Let's begin by finding the dimensions of the right circular cylinder that has maximum volume when inscribed in a sphere of radius \( R = 70 \, \text{cm} \).

### Step 1: Relationship between Cylinder and Sphere

Consider the right circular cylinder inscribed in a sphere. Let:
- \( r \) be the radius of the cylinder's base,
- \( h \) be the height of the cylinder.

For a cylinder inscribed in a sphere, the diagonal of the cylinder, which is the line joining two opposite points on the top and bottom edges of the cylinder, is equal to the diameter of the sphere.

From the Pythagorean theorem, we know that the diagonal of the cylinder can be expressed as:

\[
\sqrt{(2r)^2 + h^2} = 2R
\]

Substitute \( R = 70 \, \text{cm} \):

\[
\sqrt{(2r)^2 + h^2} = 140
\]

Squaring both sides:

\[
(2r)^2 + h^2 = 140^2
\]

\[
4r^2 + h^2 = 19600 \quad \text{(Equation 1)}
\]

### Step 2: Maximizing the Volume

The volume \( V \) of the cylinder is given by:

\[
V = \pi r^2 h
\]

We want to express \( h \) in terms of \( r \). From Equation 1:

\[
h^2 = 19600 - 4r^2
\]

Taking the square root of both sides:

\[
h = \sqrt{19600 - 4r^2}
\]

Substitute this expression for \( h \) into the volume formula:

\[
V(r) = \pi r^2 \sqrt{19600 - 4r^2}
\]

### Step 3: Finding the Critical Points

To maximize the volume, we take the derivative of \( V(r) \) with respect to \( r \) and set it equal to zero.

First, apply the product and chain rules to differentiate \( V(r) \):

\[
\frac{dV}{dr} = \frac{d}{dr} \left( \pi r^2 \sqrt{19600 - 4r^2} \right)
\]

This derivative is somewhat complex, so let me calculate it step-by-step to find the critical points.The critical point for the radius \( r \) is:

\[
r = \frac{70 \sqrt{6}}{3} \, \text{cm}
\]

### Step 4: Finding the Height \( h \)

Now, let's substitute this value of \( r \) into the equation for \( h \):

\[
h = \sqrt{19600 - 4r^2}
\]

I'll calculate the exact height.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to help walk you through the final steps manually, or assist with any part of the solution? Let me know!

Here are five related questions to expand on this topic:
1. How would the problem change if the radius of the sphere was different?
2. Can we maximize the volume of a different shape inscribed in the same sphere?
3. How would the solution change if the cylinder was not a right circular cylinder?
4. What if the cylinder were to be inscribed in a cube instead of a sphere?
5. Can you apply this method to find the dimensions of a cone inscribed in a sphere?

**Tip:** When solving optimization problems, always check the second derivative or use other tests to confirm whether a critical point is a maximum or a minimum.

Find the dimensions of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius 70 cm. What is the maximum​ volume?

Explore the steps to find the dimensions of a right circular cylinder with maximum volume inscribed in a sphere of radius 70 cm. This problem involves key concepts from geometry, calculus, and optimization, using formulas for the volume of a cylinder and relationships between the cylinder and the sphere. Suitable for college-level or advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation