A milk container is made of metal sheet in the shape of frustum of a cone whose volume is . The radii of its lower and upper circular ends are 8 cm and 20 cm respectively. Find the cost of metal sheet used in making the container at the rate of Rs 1.40 per cm2.
Upper end radius of container = R = 20 cm
Lower end radius of container =r = 8 cm
Height of the container be ‘h’ cm
As container is in shape of frustum
Volume of container = Volume of Frustum
Where R = Radius of larger or upper end and r = Radius of smaller or lower end and h = height of frustum π = 22/7
(putting the values)
⇒ 22 × h × 208 = 73216
⇒ 4576 × h = 73216
⇒ h = 73216/4576
∴ h = 16 cm
Let the slant height of the frustum be ‘ℓ’ cm
So, ℓ2 = h2 + (R – r)2
⇒ ℓ2 = 162 + (20 – 8)2
⇒ ℓ2 = 256 + (12)2
⇒ ℓ2 = 256 + 144
⇒ ℓ2 = 400
⇒ ℓ = √(400)
∴ ℓ = 20 cm
Now the Area of sheet used in making the container can be calculated by simply subtracting the area of the upper circular end from the T.S.A of frustum
⇒ T.S.A of frustum = π(R + r)ℓ + πR2 + πr2
= π(20 + 8) × 20 + π(20)2 + π(8)2
= 28 × 20π + 400π + 64π
= 560π + 464π
= 1024π →eqn1
Area of upper circular end = πR2
= π(20)2
= 400π →eqn2
T.S.A of container = 1024π – 400π (from eqn2 and 1)
⇒ T.S.A of container = 624π
⇒ Cost of metal sheet used = T.S.A of container × Rate per cm2
⇒ Cost of mtal sheet used
= 13728 × 0.2
= Rs. 2745.6
The cost of the metal sheet used is Rs. 2745.6