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andrew11 [14]
3 years ago
10

A conical cup is made from a circular piece of paper with radius 10 cm by cutting out a sector and joining the edges as shown be

low. 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.
Mathematics
1 answer:
SVEN [57.7K]3 years ago
3 0
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 <span>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 </span>θ = 9π/5 at the center which is given by
C=r\theta \\  \\ =10\times \frac{9\pi}{5}  \\  \\ =18\pi\approx56.55\ cm


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 <span>θ = 9π/5 on the circular piece of paper from which the cone was made.

</span>Recall that the circumference of a circle is given by C=2\pi r and having obtained from part a that the circumference of the circular opening is 18\pi cm.

Thus,
2\pi r=18\pi \\  \\ \Rightarrow r=9\ cm


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:
h^2+r^2=l^2
where: h is the height, r is the radius and l is the slant height.

h^2+9^2=10^2 \\  \\ \Rightarrow h^2=100-81=19 \\  \\ \Rightarrow h= \sqrt{19} \approx4.36\ cm


Part d
Recall that the volume of a cone is given by V= \frac{1}{3} \pi r^2h

Thus, the volume of the cup is given by
V= \frac{1}{3} \pi\times9^2\times \sqrt{19}  \\  \\ \approx369.7\ cm^3
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