Problem Solving

A triangle of dimensions 6, 8, and 10 is rotated about its longest side as axis. Which of the following is nearest to the volume of the solid so generated?
A. 480
B. 540
C. 630
D. 750
E. 810


Made Up

Is it 630??

I guess this will be close the the volume of a sphere with radius 5.

Please let know the correct answer and explanation.

Hi Sanju09,
Does it generate Cone?

Of Area (1/3)*3.14*r^2*h

From here I can get r = 5 since it is revolved about is longest side,
But how to get H ?

Regards,
Uva.

A 6-8-10 triangle is similar to a 3-4-5 triangle which is right angled.

Rotated around its hypotenuse will create a solid comprising 2 cones joined base to base.
See the diagram where R is proved to be 4.8.

The area of the base of each cone, B = pi * R^2 = 3.14 * 4.8^2 = 72.3 (approx.)

The volume of each cone is B * height/3, so

collectively, the total volume is V = (B * h1/3) + (B * h2/3) = B * (h1 + h2)/3 = B * 10/3

So V = 72.3 * 10/3 = 241 (approx.)

The closest (although incorrect) answer is A) 480.

sanju09 wrote:A triangle of dimensions 6, 8, and 10 is rotated about its longest side as axis. Which of the following is nearest to the volume of the solid so generated?
A. 480
B. 540
C. 630
D. 750
E. 810


Made Up

I believe the correct answer is circa 241 - please see the solution I've posted. Am I right, or have I made a fatal flaw? Nice question!

Uva@90 wrote:Hi Sanju09,
Does it generate Cone?

Of Area (1/3)*3.14*r^2*h

From here I can get r = 5 since it is revolved about is longest side,
But how to get H ?

Regards,
Uva.

See the picture I've posted (on Fri Nov 29, 2013 4:10 pm). I hope it helps

First of all, I deeply regret for supplying the wrong answer choices, which were meant for some other question. Please consider the following answer choices for this question:

A. 150
B. 180
C. 210
D. 250
E. 270

Please draw on your own according to the following explanation:

∆ABC has AB = 6, BC = 10, and CA = 8, and consequently it is right angled at A. Mathsbuddy did justice to the situation that I described in the question. Now drop perpendicular from A on BC to meet BC in D. This AD would serve as the common radius in the resulting solid so generated. Area of ∆ABC is ½ AB.AC = ½ BC.AD = 24, hence AD = 4.8; for approximation purposes we'll later on take AD = 5. Revising properties of right triangles anybody could come to believe that the three triangles so formed are all similar. Set up the following similarity relation

AB/AC = BD/AD = AD/DC= 6/8 (*This step is actually not very necessary, this we'd realise towards the end of our calculations.)

Since AD = 4.8, hence 4.8/DC = 6/8 or DC = 6.4 and hence BD = 3.6, this solves Uva@90's 'H' issue. Yes, it does generate two cones surmounted over one another as correctly and beautifully pictured by Mathsbuddy.

The collective volume = 1/3 π (5)^2 (3.6) + 1/3 π (5)^2 (6.4) [Let's take π = 3 only]

This would result in

The collective volume = (5)^2 (3.6) + (5)^2 (6.4) = (5) ^2 (*3.6 + *6.4) = [spoiler](25) (*10) = 250. It's D.[/spoiler]