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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 6, Problem 3P

(a)

To determine

**To show:** The volume of a segment of a sphere

Expert Solution

The volume of the segment of a sphere is

**Given information:**

The segment of a sphere with radius *r* and height *h*.

Consider that the Equation of circle as

Rearrange Equation (1).

The dimensions of the sphere as shown in Figure 1.

Refer to Figure 1.

Consider that the sphere is obtained by rotating the circle *y*-axis.

The region lies between

**Calculation:**

The expression to find the volume of the segment of a sphere as shown below:

Find the area of the segment of a sphere as shown below.

Substitute *a*, *r* for *b*, and

Therefore, the volume of the segment of a sphere is

(b)

To determine

**To calculate:** The value of *x* using Newton’s method.

Expert Solution

The value of *x* by using Newton’s method is 0.2235.

**Given information:**

A sphere of radius 1 is sliced by a plane at a distance *x* from the center.

The volume of one segment is twice the volume of the other.

The Answer of the equation *x*.

**Calculation:**

Find the volume of the sphere as shown below.

Substitute 1 for *r* in Equation (3).

Consider the smaller segment has height

Refer to part (a).

Volume of the segment of a sphere as shown below:

Substitute 1 for *r* and *h* in Equation (4).

This volume must be

Substitute *V* in the above Equation.

Apply Newton’s method as shown below.

Consider

Differentiate both sides of the Equation.

Substitute

Consider

Substitute 1 for *n* in Equation (6).

Substitute 2 for *n* in Equation (6).

Substitute 2 for *n* in Equation (6).

Therefore, the value of *x* by using Newton’s method is 0.2235.

(c)

To determine

**To calculate:** sinking depth of the sphere.

Expert Solution

The sinking depth of the sphere is 0.6736 m.

**Given information:**

Wooden sphere of radius as 0.5 m.

Specific gravity of the wooden sphere is 0.75.

The depth *x* to which a floating sphere of radius *r* sinks in water is a root of the equation as follows:

**Calculation:**

Show the root of the equation as shown below.

Substitute 0.5 m for *r* and 0.75 for *s* in Equation (7).

Consider

Differentiate both sides of the Equation.

Apply Newton’s method.

Substitute

Consider

Substitute 1 for *n* in Equation (8).

Substitute 2 for *n* in Equation (8).

Substitute 3 for *n* in Equation (8).

Approximately the depth should be

Therefore, the sinking depth of the sphere is 0.6736 m.

(d) i)

To determine

**To calculate:** The speed of rising the water level in the bowl from instant of water level at 3 inches deep.

Expert Solution

The speed of rising the water level in the bowl from instant of water level at 3 inches deep is

**Given information:**

The shape of the bowl is hemisphere.

The radius of hemispherical bowl is 5 inches.

Water is running into the bowl at the rate of

Depth of water is 3 inches.

**Calculation:**

Show the volume of the segment of a sphere as follows:

Differentiate both sides of the Equation with respect to time *t*.

Find the speed of the water level in the bowl rising at the instant as shown below.

Substitute 5 inches for *r*, *h* in Equation (9).

Therefore, the speed of rising the water level in the bowl from instant of water level at 3 inches deep is

ii)

To determine

**To calculate:** The time taken to fill the bowl from 4 inch level of water.

Expert Solution

The time taken to fill the bowl from 4 inch level of water is

**Given information:**

The radius of hemispherical bowl is 5 inches.

Water is running into the bowl at the rate of

Depth of water is 4 inches.

**Calculation:**

Find the volume of the water required to fill the bowl from the instant that the water is 4 in. depth as follows:

Substitute 5 inches for *r* and 4 inches for *h* in the Equation.

Find the time required to fill the bowl as shown below.

Therefore, The time taken to fill the bowl from 4 inch level of water is