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Calculus IIChapter 6: 3-Dimensional SpaceSection 6-1

The 3-D Coordinate System — Question 8

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Question 8

A rectangular box has one vertex at the origin and the opposite vertex P=(a,b,c)P=(a,b,c) in the first octant. Its space diagonal has length 13, its xyxy-face diagonal has length 5, and the yy-edge is longer than the xx-edge. If the side lengths are positive integers, determine PP.

Original worksheet page 1: question and worked solution for 6-1-008
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Question 8 – Solution

Strategy Use the face diagonal to determine two integer side lengths, then use the space diagonal for the third.

See the diagram in the original worksheet below.

Base dimensions The positive integer side lengths satisfying a2+b2=25a^2+b^2=25 are 3 and 4. Since the yy-edge is longer than the xx-edge, a=3a=3 and b=4b=4.

Third dimension The space diagonal gives a2+b2+c2=169⇒25+c2=169,a^2+b^2+c^2=169\quad\Longrightarrow\quad 25+c^2=169, so c=12c=12. Therefore P=(3,4,12)\boxed{P=(3,4,12)}.

Verification The face diagonal is 32+42=5\sqrt{3^2+4^2}=5 and the space diagonal is 32+42+122=13\sqrt{3^2+4^2+12^2}=13.

Original worksheet page 2: question and worked solution for 6-1-008

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