perplexus.info :: Shapes : Hypotenuse from Cone

Hypotenuse from Cone (Posted on 2016-01-22)

PQR is a right angled triangle, where PR is the hypotenuse.
When this triangle is rotated about PQ, the volume of the cone produced is 800 cubic inches.
When the triangle is rotated about QR, the volume of the cone produced is 1920 cubic inches.

Determine (in inches) the hypotenuse of the triangle.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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Solution

| Comment 1 of 2

The volume of a cone is V=(pi/3)*r^2*h. Then:

Volume of PQ rotation: 800 = (pi/3)*QR^2*PQ

Volume of QR rotation: 1920 = (pi/3)*PQ^2*QR

A pretty straightforward system that resolves to:

PQ = 24/cbrt(pi)

QR = 10/cbrt(pi)

Then just apply Pythagorean Theorem to get:

QR = sqrt[(24/cbrt(pi))^2 + (10/cbrt(pi))^2] = 26/cbrt(pi)

Posted by Brian Smith on 2016-01-22 09:51:40

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