from a handpicked tutor in LIVE 1-to-1 classes

# 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 be drawn into a wire of diameter 1/16 cm, find the length of the wire

**Solution:**

A figure is drawn below to visualize the cross-section of a cone.

As mentioned above, a metallic right circular cone is cut into two parts at the middle of its height by a plane parallel to its base.

To get the values of radii of both ends of the frustum formed, compare ΔADB and ΔADC

Since the frustum obtained is drawn into the wire which will be cylindrical in shape, the volume of the wire will be the same as the volume of the frustum of the cone.

Therefore, Volume of the wire = Volume of frustum of the cone

We will find the volume of the frustum by using formula;

Volume of frustum of a cone = 1/3 πh (r₁^{2} + r₂^{2} + r₁r₂), where r₁, r₂_{,} and h are the radii and height of the frustum of the cone respectively.

We will find the volume of the wire by using formula;

Volume of cylinder = πr^{2}h, where r and h are radius and height of the cylinder

In ΔABC , EF parallel to BC and

AD = 20 cm

AG = 10 cm

∠BAC = 60°

To get the values of radii of both ends of the frustum formed compare ADB and ADC

AD = AD (common)

AB = AC (Slant height )

∠ADB = ∠ADC = 90 (Right circular cone)

ΔADB ≅ ΔADC (RHS criterion of congruency)

∠BAD = ∠DAC (CPCT)

Then,

∠BAD = ∠DAC

∠BAD = 1/2 × 60° = 30°

In ΔADB

BD/AD = tan 30°

BD = AD tan 30°

BD = 20 cm × 1/√3

BD = 20√3/3 cm

Similarly, in ΔAEG

EG/AG = tan 30°

EG = AG tan 30°

EG = 10 cm × 1/√3

EG = 10√3/3 cm

Height of the frustum of the cone, h = 10 cm

Radius of lower end, r₁ = (20√3) / 3 cm

Radius of upper end, r₂ = (10√3) / 3 cm

Diameter of the cylindrical wire, d = 1 / 16 cm

Radius of the cylindrical wire, r = 1 / 2 × 1 / 16 cm = 1 / 32 cm

Let the length of the wire be H

Since the frustum is drawn into wire

Volume of the cylindrical wire = Volume of frustum of the cone

πr^{2}H = 1/3 πh(r₁^{2} + r₂^{2} + r₁r₂)

H = [h(r₁^{2} + r₂^{2} + r₁r₂)]/ 3r^{2}

= [ 10 × {(20√3) /3)^{2} + (10√3/ 3)^{2} + (20√3) /3 × (10√3) /3] / 3 × (1/32)^{2}

= [10 × {400/3 + 100/3 + 200/3}] / 3 × (1/1024)

= 10240 × 700 / 9

= 7168000 / 9

= 796444.44 cm

= 7964.4 m

Thus the length of the wire is 7964.4 m.

**☛ Check: **Class 10 Maths NCERT Solutions Chapter 13

**Video Solution:**

## 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 be drawn into a wire of diameter 1/16 cm, find the length of the wire

NCERT Solutions for Class 10 Maths Chapter 13 Exercise 13.4 Question 5

**Summary:**

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 be drawn into a wire of diameter 1/16 cm, the length of the wire 7964.4 m.

**☛ Related Questions:**

- 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.
- The slant height of a frustum of a cone is 4 cm and the perimeters (circumference) of its circular ends are 18 cm and 6 cm. Find the curved surface area of the frustum.
- A fez, the cap used by the Turks, is shaped like the frustum of a cone (see Fig. 13.24). 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 used for making it.
- 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 ₹ 20 per litre. Also find the cost of metal sheet used to make the container, if it costs ₹ 8 per 100 cm².

visual curriculum