Aconical cup is made from a circular piece of paper with radius 10 cm by cutting out a sector and joining the edges as shown below. suppose θ = 9π/5. (a) find the circumference c of the opening of the cup. (b) find the radius r of the opening of the cup. c) find the height h of the cup. (d) find the volume v of the cup.

Part a:

The opening of the cup is the circular base of the cup which has a circumference equal to the length of the arc formed by angle θ = 9π/5 on the circular piece of paper from which the cone was made.

Thus, the circumference of the circle = Length of the arc formed by angle θ = 9π/5 at the center which is given by


Part b:
The opening of the cup is the circular base of the cup which has a circumference equal to the length of the arc formed by angle θ = 9π/5 on the circular piece of paper from which the cone was made.

Recall that the circumference of a circle is given by and having obtained from part a that the circumference of the circular opening is cm.

Thus,


Part c:
The height of the cup can be obtained by noticing that the radius, height and the slant height of the cup forms a right triangle with the height and the radius as the legs and the slant height as the hypothenus.

Using pythagoras theorem, the height of the cup is obtained as follows:

where: h is the height, r is the radius and l is the slant height.



Part d
Recall that the volume of a cone is given by

Thus, the volume of the cup is given by

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