Maxima and Minima question (1 Viewer)

kevda1st · May 4, 2009

Find the Volume of the largest cone that can be inscribed in a sphere of radius 9cm.

Answers were not given.

Thanks in advance

jet · May 4, 2009

Well, you can see from the diagram that the radius of the sphere is equal to $\sqrt{81 - x^2}$
Hence, the volume of the cone is equal to $V = \frac{1}{3}\pi \left (\sqrt{81 - x^2} \right )^2 \left ( x + 9 \right )$
x + 9 is the height of the cone btw, as the length from the centre of the sphere to the apex of the cone is just the radius.
$\therefore V = \frac{\pi}{3}(-x^3 - 9x^2 + 81x + 729) \\ \frac{dV}{dx} = \frac{\pi}{3}(-3x^2 - 18x + 81) = 0 $ for stationary points$ \\ x^2 + 6x - 27 = 0 \\ (x + 9)(x - 6) = 0 \\ \thereofre x = -9 $ OR $ x = 6 \\ \frac{d^2 V}{dx^2} = \frac{\pi}{3}(-6x - 18) \\ $When $ x = -9, \frac{d^2 V}{dx^2} > 0, \therefore $ minimum$ \\ x = 6, \frac{d^2 V}{dx^2} < 0 \therefore $ maximum$ \\ $Hence, the largest volume of the cone is $ 225\pi $ after substitution back into the formula$$

gurmies · May 4, 2009

jetblack2007 said:
Well, you can see from the diagram that the radius of the sphere is equal to $\sqrt{81 - x^2}$
Hence, the volume of the cone is equal to $V = \frac{1}{3}\pi \left (\sqrt{81 - x^2} \right )^2 \left ( x + 9 \right )$
x + 9 is the height of the cone btw, as the length from the centre of the sphere to the apex of the cone is just the radius.
$\therefore V = \frac{\pi}{3}(-x^3 - 9x^2 + 81x + 729) \\ \frac{dV}{dx} = \frac{\pi}{3}(-3x^2 - 18x + 81) = 0 $ for stationary points$ \\ x^2 + 6x - 27 = 0 \\ (x + 9)(x - 6) = 0 \\ \thereofre x = -9 $ OR $ x = 6 \\ \frac{d^2 V}{dx^2} = \frac{\pi}{3}(-6x - 18) \\ $When $ x = -9, \frac{d^2 V}{dx^2} > 0, \therefore $ minimum$ \\ x = 6, \frac{d^2 V}{dx^2} < 0 \therefore $ maximum$ \\ $Hence, the largest volume of the cone is $ 225\pi $ after substitution back into the formula$$

Very well done.

ghostman_on_3rd · May 4, 2009

hmm, Jetblack" your factorising:

" 0=3(x^2+6x-27)"
I believe it should factorise into: 0 = (x+9)(x-3)

instead of "0 = (x+9) (x- 6)"

new answer = 288pi cm^3 ?

gurmies · May 4, 2009

ghostman_on_3rd said:
hmm, Jetblack" your factorising:

" 0=3(x^2+6x-27)"
I believe it should factorise into: 0 = (x+9)(x-3)

instead of "0 = (x+9) (x-6)"
_________________________
new answer = 288pi cm^3

Well spotted.

ghostman_on_3rd · May 4, 2009

gurmies said:
Well spotted.

Thanks

Maxima and Minima question (1 Viewer)

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