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Hello and welcome to this video solution of numerate.
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Here you have to imagine three planets all of which have same mass.
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Planet a is solid iron, planet b is solid water, ice and planet c is a gas giant.
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And with an average density of one half of the liquid water.
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So here what we see is that the mass are constant for the planet.
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Planet a is made of iron, b is solid water, ice we can say, c is gas.
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Here it's given that the density of the gas rho g let's say and this is a density of water and this is density of iron.
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And it's given that the density of water is twice that of the gas or the gas is half water i'm taking on the other way around.
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And it's given that what are the ratios of the planet radius and it's given that the iron is 7 .9 times the density of liquid water.
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So it's 7 .9 times rho w.
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Now you can plug in you can convert everything in terms of the density of gas which is 7 .9 times of 2 times of rho g.
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This is equal to 15 .8 rho g.
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So here what we have is the masses are equal.
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Mass of a is equal to mass of b that is equal to mass of c.
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Now mass is given by the density times the volume.
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Density of a times the volume of a.
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Density of b times volume of b.
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And density of c times volume of c.
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Now density of a is for that iron which is 15 .8 rho g.
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Rho g and the volume will be four third pi r a cube.
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You consider this to be a sphere.
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Similarly rho b is two times rho g times four third pi r b cube.
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That is equal to rho g times four third pi r c cube.
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Right.
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Four third pi can be cancelled.
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Rho g can be cancelled from all sides.
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That by rho g.
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Right...