The radius of the base of the cylinder is 2x cm and the height of the cylinder is h cm.
The radius of the sphere is 3x cm.
The volume of the cylinder is equal to the volume of the sphere.
Express h in terms of x. Give your answer in its simplest form.

Answer:

h = 9x

Step-by-step explanation:

Given: i. for the cylinder, base radius = 2x cm, height = h cm

           ii. for the sphere, radius = 3x cm

           iii.  volume of the cylinder = volume of the sphere

volume of a cylinder is given as;

volume = [tex]\frac{1}{3}[/tex][tex]\pi[/tex][tex]r^{2}[/tex]h

where: r is its base radius and h the height

volume of the given cylinder = [tex]\frac{1}{3}[/tex][tex]\pi[/tex] x [tex](2x)^{2}[/tex] x h

                                              = [tex]\frac{1}{3}[/tex][tex]\pi[/tex] x 4[tex]x^{2}[/tex] x h

                                              = [tex]\frac{4}{3}[/tex][tex]\pi[/tex][tex]x^{2}[/tex] h

volume of a sphere = [tex]\frac{4}{3}[/tex][tex]\pi[/tex][tex]r^{3}[/tex]

where r is the radius.

volume of the given sphere =  [tex]\frac{4}{3}[/tex][tex]\pi[/tex] x [tex](3x)^{3}[/tex]

                                         =  [tex]\frac{4}{3}[/tex][tex]\pi[/tex] x 9 [tex]x^{3}[/tex]

                                         = 12[tex]\pi[/tex][tex]x^{3}[/tex]

Since,

volume of the cylinder = volume of the sphere

Then we have;

[tex]\frac{4}{3}[/tex][tex]\pi[/tex][tex]x^{2}[/tex] h = 12[tex]\pi[/tex][tex]x^{3}[/tex]

4[tex]\pi[/tex][tex]x^{2}[/tex] h = 36[tex]\pi[/tex][tex]x^{3}[/tex]

subtract [tex]x^{2}[/tex] from both sides

4[tex]\pi[/tex]h = 36[tex]\pi[/tex]x

divide both sides by 4[tex]\pi[/tex]

h = [tex]\frac{36\pi x}{4\pi }[/tex]

  = 9x

h = 9x

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