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In the diagram below, a right circular cone with a radius of 3 inches has a slant height of 5 inches, and a right cylinder with a radius of 4 inches has a height of 6 inches.

Determine and state the number of full cones of water needed to completely fill the cylinder with water.

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Number of full cones of water needed to completely fill the cylinder with water, N = Volume of Cylinder, V_{1} / Volume of Cone, V_{2}.

That is, N = V_{1}/V_{2}

V_{1} = πr_{1}^{2}h_{1}

where r_{1} and h_{1} are the radius and height of the cylinder respectively.

V_{1} = π(4)^{2}(6)

= 96π

V_{2} = 1/3 πr_{2}^{2}h_{2}

Where r_{2} and h_{2} are radius and vertical height of the cone respectively.

From pythagorean's theorem, h_{2} = 5^{2} - 3^{2}

h_{2}^{2} = 25 - 9

h_{2}^{2} = 16

Take square root of both sides

h_{2} = 4

Thus, V_{2} = 1/3 π(3)^{2}(4)

= 12π

Therefore, N = 96π / 12π

= 8

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