What if the volume of a cylinder as a function of it's height/radius? Full question in the description box below.

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1 Answer
Jan 4, 2018

V_1(h) = 1/16 h(144-h^2)" cm"^3" " for h in [0, 12]

V_2(r) = 2pir^2 sqrt(36-r^2)" cm"^3" " for r in [0, 6]

Explanation:

If we take a vertical slice through the centre of the sphere and cylinder, then it looks like this:

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Where r is the radius of the cylinder, R the radius of the sphere and h the height of the cylinder.

Then the volume of the cylinder is the area of its base, multiplied by the height. That is:

V = h pi r^2

Note that the radius r varies as the height h varies, always satisfying Pythagoras formula:

(2R)^2 = h^2+(2r)^2

That is:

4R^2 = h^2+4r^2

Hence we can write r in terms of h or h in terms of r:

r = 1/4 sqrt(4R^2-h^2)

h = sqrt(4R^2-4r^2) = 2sqrt(R^2-r^2)

We can substitute these formulae into our prior formula for the volume of the cylinder to find:

V_1(h) = h pi r^2 = h pi (1/4 sqrt(4R^2-h^2))^2 = 1/16 h(4R^2-h^2)

V_2(r) = h pi r^2 = (2sqrt(R^2-r^2)) pi r^2 = 2pir^2 sqrt(R^2-r^2)

Finally, substituting R=6 (cm) we get:

V_1(h) = 1/16 h(144-h^2)" cm"^3" " for h in [0, 12]

V_2(r) = 2pir^2 sqrt(36-r^2)" cm"^3" " for r in [0, 6]