Question

the radii of the bases of two right circular solid cones of same height are r1 and r2 respectively.the cones are melted and recasted into a solid sphere of radius R.Show that the height of each cone is given by

h=4R³÷²+²

Answer 1

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Answer 2

To prove that $h=\frac{4 R^{3}}{r_{1}^{2}+r_{2}^{2}}$

Step-by-step explanation:

Given data:

Radius of two cones are $r_1$ and $r_2$.

Height of the two cones are same which is h.

Volume of cone 1 = $\frac{1}{3} \times\pi \times r_1 ^2 \times h$

Volume of cone 2 = $\frac{1}{3} \times\pi \times r_2^2 \times h$

Let R be the radius of the sphere

Volume of the sphere = $\frac{4}{3} \times \pi\times R^{3}$

Given that,

Volume of the sphere = volume of cone 1 + volume of cone 2

$\frac{4}{3} \times \pi\times R^{3}=\frac{1}{3} \times\pi \times r_1^2 \times h+\frac{1}{3} \times\pi \times r_2^2 \times h$

simplifying the expression.

$4 \times \pi\times R^{3}=\pi \times r_1^2 \times h+\pi \times r_2^2 \times h$

$4 \times \pi\times R^{3}=\pi h (r_1^2 + r_2^2 )$

Cancel the common terms in both sides, we get

$4 \times R^{3}=h (r_1^2 + r_2^2 )$

Divide by $r_1^2 + r_2^2$ on both sides, we get

$h=\frac{4 R^{3}}{r_{1}^{2}+r_{2}^{2}}$

Hence proved.

To learn more...

1. Hollow sphere of internal and external diameter 4 cm and 8 cm respectively is melted into a cone of base diameter 8 centimetre find the height of the cone

https://brainly.in/question/2084958

2. A Cone is 8.4 cm high and the radius of its base is 2.1cm. It is melted and recast into a sphere. Find the radius of sphere formed?

https://brainly.in/question/320891