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Question 2

Number and Inequalities

1.A cylindrical vessel with one end open is made from a given piece of material. Show that its volume is greatest if the height and radius are equal.

2. a,b,c are non zero rational numbers. Show that if:

³√a + ³√b + ³√c = 0

And if none of ³√a , ³√b , ³√c is rational then ³√a , ³√b , ³√c are rational multiples of the same irrational number.

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Question 2

Number and Inequalities

1.A cylindrical vessel with one end open is made from a given piece of material. Show that its volume is greatest if the height and radius are equal.

As the cylinder is made from a given piece of material, the surface area of the cylinder remains constant. Let the surface area of the cylinder is A, and is constant.
Let the cylinder has the radius r and height h.
As the cylinder is open at one end, the total surface area is given by A = (pie)r^2 + 2(pie)rh...(1)
Solving this for 'h' we get:
h = A - (pie)r^2/2(pie)r...(2)
We know that the volume of the cylinder is given by V = (pie)r^2 h...(3)
Using the equation (2), we can write equation (3) as
(please see the attached file)

Differentiating with respect to r., we ...

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