1: A copper wire having a diameter of 3mm, is wound about a cylinder whose length is 12 cm, and diameter 10 cm, so as to cover the curved surface of the cylinder. Calculate the length and mass of the wire, assuming the density of copper to be 8.88 g per cm^{3}.

Ans.-Given, For copper wire: Diameter is = 3 mm = 0.3 cm

For cylinder; length h is = 12 cm, d = 10 cm

Density of copper is = 8.88 gm cm^{3}

So now the curved surface area of cylinder can be calculated as follows:

We know that-: Length =

=

= 1256cm

Similarly, volume of wire can be calculated as follows:

V =

=

Hence, mass can be calculated as-: Mass = Density*Volume

2: A right angled triangle, whose sides with 3 cm and 4 cm (other than hypotenuse) is made to revolve about its hypotenuse. Calculate the volume and surface area of the double cone so formed.

Ans.- In the given triangle ABC;

Or,

Or, AC=5CM

In triangle ABC and triangle BDC;

Hence,

So, now we get following equations:

Or,

In triangle BDC;

Or,

=

Upper Cone: r = 2.4 cm, l = 3 cm, h = 1.8 cm

We know that, volume of cone is

=

Now, Curved surface area of cone

=

=

Similarly, for lower cone: r = 2.4 cm, l = 4 cm, h = 3.2 cm

As we know that, volume of cone is

=

=

Now, Curved surface area of cone is

=

=

Total volume = 19.29216 + 10.85184=30.144cm

3: A cistern measuring 150 cm x 120 cm x 110 cm has 129600

Ans.- Here volume of cistern is = length x width x depth

= 150 x 120 x 110

= 1980000

Vacant space = Volume of cistern – Volume of water

= 1980000 – 129600 = 1850400

Volume of brick = length x width x height

= 22.5 x 7.5 x 6.5

Since the brick absorbs one seventeenth its volume hence the remaining volume will be equal to 16/17 the volume of brick

Remaining volume is

=

=

=

5: An funnel made of a metallic sheet consisting of a 10 cm long cylindrical portion attached to a frustum of a cone. If the total height and the diameter of the cylindrical portion are 22 cm and 8 cm respectively and diameter of the top of the funnel is 18 cm, calculate the area of the metallic sheet required to make the funnel.

Ans.- Curved surface area of cylinder

Slant height of frustum can be calculated as follows:

Curved surface area of frustum

=

=