Surface Areas and Volumes
Exercise 13.4
Question 1:
A drinking glass is in the shape of a frustum of a cone of height 14 cm. The diameters of its two circular ends are 4 cm and 2 cm. Find the capacity of the glass.
Radius (r1) of upper base of glass = 4/2 = 2cm
Radius (r2) of lower base of glass =2/2 = 1cm
Capacity of glass = Volume of frustum of cone
Therefore, the capacity of the glass is
.
Question 2:
Perimeter of upper circular end of frustum = 18
2∏r1 =18
Perimeter of lower end of frustum = 6 cm
2∏r2 = 6
Slant height (l) of frustum = 4
CSA of frustum = ∏ (r1 + r2) l
Therefore, the curved surface area of the frustum is 48 cm2.
Question 3:
A fez, the cap used by the Turks, is shaped like the frustum of a cone (see the figure given below). If its radius on the open side is 10 cm, radius at the upper base is 4 cm and its slant height is 15 cm, find the area of material use for making it.
Radius (r2) at upper circular end = 4 cm
Radius (r1) at lower circular end = 10 cm
Slant height (l) of frustum = 15 cm
Area of material used for making the fez = CSA of frustum + Area of upper circular end
= ∏ (10 + 4) 15 + ∏ (4)2
= ∏ (14) 15 + 16 ∏
Therefore, the area of material used for making it is
.
Question 4:
A container, opened from the top and made up of a metal sheet, is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of the milk which can completely fill the container, at the rate of Rs.20 per litre. Also find the cost of metal sheet used to make the container, if it costs Rs.8 per 100 cm2. [Take π = 3.14]
Radius (r1) of upper end of container = 20 cm
Radius (r2) of lower end of container = 8 cm
Height (h) of container = 16 cm
Slant height (l) of frustum =
Capacity of container = Volume of frustum
Cost of 1 litre milk = Rs 20
Cost of 10.45 litre milk = 10.45 × 20
= Rs 209
Area of metal sheet used to make the container
= π (20 + 8) 20 + π (8)2
= 560 π + 64 π = 624 π cm2
Cost of 100 cm2 metal sheet = Rs 8
Therefore, the cost of the milk which can completely fill the container is Rs 209 and the cost of metal sheet used to make the container is Rs 156.75.
Question 5:
A metallic right circular cone 20 cm high and whose vertical angle is 60° is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained is drawn into a wire of diameter 1/16 cm, find the length of the wire.
In ΔAEG,
In ΔABD,
Radius (r1) of upper end of frustum =
cmRadius (r2) of lower end of container =
Height (h) of container = 10 cm
Volume of frustum
Radius (r) of wire =
Let the length of wire be l.
Volume of wire = Area of cross-section × Length
= (Ï€r2) (l)
Volume of frustum = Volume of wire
