Chapter 14.5, Problem 35E

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11th Edition

Ron Larson + 1 other

ISBN: 9781337275347

Textbook Problem

Product DesignA company produces a spherical object of radius 25 centimeters. A hole of radius 4 centimeters is drilled through the center of the object.

(a) Find the volume of the object.

(b) Find the outer surface area of the object.

**(a)**

To determine

**To calculate:** The volume of the object where the radius of the spherical object is given to be 25 centimetres and it is mentioned that a hole of radius 4 centimetres is drilled through the centre of the same object.

Explanation

**Given:**

The radius of the given spherical object is 25 centimetres. 4 centimetres is the radius of the hole that is drilled through centre of the object.

**Formula Used:**

The volume can be found by performing double integration on the surface within the limits.

The equation of sphere can be written as

The equation of circle can be written as

Integration formula to be used is

**Calculation:**

As given in the question, the object is a sphere of radius 25 centimetres; a hole is drilled through centre of the object and the hole is of radius 4 centimetres.

The region is

Then,

On doing so, the net region obtained appears in the *x-y* plane in the first quadrant, which can be described as shown below.

For the region being discussed, the limits for both x and y shall be converted to polar points.

Thus, the region being discussed is bounded by

The equation of the region can be written as:

Substitute the equation obtained above in the formula of volume.

In order to account for all the hemispheres, the surface is multiplied by 8.

Take

**(b)**

To determine

**To calculate:** The outer surface area of the given object, which is spherical in shape with a radius of 25 centimetres; it also has a hole of radius 4 centimetres is drilled through centre of it.

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