Chapter 3: Q. 30 (page 288)

Use optimization techniques to answer the questions in Exercises 25–30.
Find the volume of the largest cylinder that fits inside a sphere of radius $10$ .

Short Answer

Expert verified

The volume of the largest cylinder is $V = \frac{4000 π}{3 \sqrt{3}}$ .

Step by step solution

Step 1. Given Information.

Radius of circle is $10$ .

Step 2. Form an equation.

From the given information,

$r^{2} = R^{2} - {(\frac{h}{2})}^{2} V = {πr}^{2} h = π (R^{2} - {(\frac{h}{2})}^{2}) h$

Step 3. Differentiate with respect to h.

On differentiating,

$\frac{d V}{d h} = π ({hR}^{2} - \frac{h^{2}}{4}) π ({hR}^{2} - \frac{h^{2}}{4}) = 0$

Step 4. Find the maximum volume.

The second derivative of the volume with respect to h is negative is $h > 0$ such that the volume is maximal at $h = h_{0}$ .

$V = \frac{4 {πR}^{3}}{3 \sqrt{3}} = \frac{4 π {(10)}^{3}}{3 \sqrt{3}} V = \frac{4000 π}{3 \sqrt{3}}$

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Use optimization techniques to answer the questions in Exercises 25–30.
Find the volume of the largest cylinder that fits inside a sphere of radius $10$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Form an equation.

Step 3. Differentiate with respect to h.

Step 4. Find the maximum volume.

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Use optimization techniques to answer the questions in Exercises 25–30.Find the volume of the largest cylinder that fits inside a sphere of radius 10.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Form an equation.

Step 3. Differentiate with respect to h.

Step 4. Find the maximum volume.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Calculus

Discrete Mathematics

Logic and Functions

Decision Maths

Statistics

Study anywhere. Anytime. Across all devices.

Use optimization techniques to answer the questions in Exercises 25–30.
Find the volume of the largest cylinder that fits inside a sphere of radius $10$ .