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High School Math Calculus PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)

Tofind: The standard form of the equation of sphere with the endpoints of a diameter as ( 3 , 0 , 0 ) and ( 0 , 0 , 6 ) .

Tofind: The standard form of the equation of sphere with the endpoints of a diameter as ( 3 , 0 , 0 ) and ( 0 , 0 , 6 ) .

PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)

PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)

8th Edition

ISBN: 9781337904285

Author: Larson

Publisher: CENGAGE L

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Question

Chapter 10.1, Problem 63E

To determine

Tofind:The standard form of the equation of sphere with the endpoints of a diameter as (3,0,0) and (0,0,6) .

Expert Solution & Answer

Answer to Problem 63E

The standard form of the equation of sphere with endpoints of the diameter as (3,0,0) and (0,0,6) is (x−32)2+y2+(z−3)2=(352)2 .

Explanation of Solution

Given information:

The vertices of endpoints of a diameter are (3,0,0) and (0,0,6) .

Calculation:

The general standard form of the equation of sphere is (x−h)2+(y−k)2+(z−j)2=r2 with center (h,k,j) and r radius.

The center of the sphere is the midpoint of the vertices of endpoints of diameter.

Use the formula for midpoint.

Midpoint=(x1+x22,y1+y22,z1+z22)

Substitute 3 , 0 , 0 for x1 , y1 , z1 and 0 , 0 , 6 for x2 , y2 , z2 in the midpoint formula.

Midpoint=(3+02,0+02,0+62)=(32,0,62)=(32,0,3)

So, the center of the sphere is (32,0,3) .

Use the distance formula for the length of diameter.

Distance=(x2−x1)2+(y2−y1)2+(z2−z1)2

Substitute 3 , 0 , 0 for x1 , y1 , z1 and 0 , 0 , 6 for x2 , y2 , z2 in the distance formula.

Distance=(0−3)2+(0−0)2+(6−0)2=(−3)2+0+62=9+0+36=45=35

So, the length of the diameter is 35 .

The radius of sphere is half the length of the diameter.

r=352

Substitute 32 , 0 , 3 for h , k , j and 352 for r in the general equation of sphere.

(x−h)2+(y−k)2+(z−j)2=r2(x−32)2+(y−0)2+(z−3)2=(352)2(x−32)2+y2+(z−3)2=(352)2

Therefore, the standard form of the equation of the sphere with endpoints of the diameter as (3,0,0) and (0,0,6) is (x−32)2+y2+(z−3)2=(352)2 .

Chapter 10.1, Problem 62E

Chapter 10.1, Problem 64E

Chapter 10 Solutions

PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)

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