Question

A bucket is in the form of a frustum of a cone with a capacity of 12308.8 cm³ of water. The radii of the top and bottom circular ends are 20 cm and 12 cm respectively. Find the height of the bucket and the area of the metal sheet used in its making. (Use π = 3.14).

Answer 1

Answer:

Height of the bucket (h) = 15 cm  and

Area of a metal sheet used in making a bucket = 2160.32 cm²

Step-by-step explanation:

SOLUTION :  

Given :  

Radius of top circular ends ,R = 20 cm

Radius of bottom circular end of bucket ,r = 12 cm

Volume / Capacity of a bucket = 12308.8 cm³

Let the height of bucket be ‘h’.

Volume  of of a bucket = π/3  (R² + r² + Rr) h

= ⅓ Π(20² + 12² + 20 × 12)h

= ⅓ × π(400 + 144 + 240)h

= ⅓ π (784)h

= (22/7) × ⅓ × 784 × h

= (22 × 112 × h)/3

Volume  of of a bucket = 2464h/3

12308.8 = (22 × 112 × h)/3

h = (12308.8 × 3)/(22 × 112)

h = 36,926.4/2464

h = 14.98 ≈ 15 (approximately)

h = 15 cm

Height of the bucket (h) = 15 cm

Slant height of a frustum , l = √(R - r)² + h²

l = √(20 - 12)² + 15²

l = √8² + 15² = √64 + 225 = √289

l  = 17 cm

Slant height of the bucket ,l  =17 cm

Area of a metal sheet used in making a bucket = π(r1+ r2)l + πr²

= π(20 + 12)17 + π ×12²

= π(32 × 17) + 144π

= π(544 + 144) = 688 π  

= 688 × 3.14 = 15136/7

= 2160.32 cm²

Hence, the Area of a metal sheet used in making a bucket = 2160.32 cm²

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Answer 2

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Height of the bucket (h) = 15 cm

 and

Area of a metal sheet used in making a bucket = 2160.32 cm²

step-by-step explanation:

Given,

Radius of top circular ends ,R = 20 cm

and,

Radius of bottom circular end of bucket ,r = 12 cm

also,

Volume / Capacity of a bucket = 12308.8 cm³

now,

Let the height of bucket be ‘h’.

thus,

Volume  of of a bucket = π/3  (R² + r² + Rr) h

= ⅓ Π(20² + 12² + 20 × 12)h

= ⅓ × π(400 + 144 + 240)h

= ⅓ π (784)h

= (22/7) × ⅓ × 784 × h

= (22 × 112 × h)/3

Volume  of of a bucket = 2464h/3

=> 12308.8 = (22 × 112 × h)/3

=> h = (12308.8 × 3)/(22 × 112)

=>h = 36,926.4/2464

=> h = 14.98 ≈ 15 (approximately)

=> h = 15 cm

so,

Height of the bucket (h) = 15 cm

now,

Slant height of a frustum , l = √(R - r)² + h²

=> l = √(20 - 12)² + 15²

=> l = √8² + 15² = √64 + 225 = √289

=> l  = 17 cm

so,

Slant height of the bucket ,l  =17 cm

now,

Area of a metal sheet used in making a bucket

= π(r1+ r2)l + πr²

= π(20 + 12)17 + π ×12²

= π(32 × 17) + 144π

= π(544 + 144) = 688 π  

= 688 × 3.14 = 15136/7

= 2160.32 cm²

Hence,

the Area of a metal sheet used in making a bucket = 2160.32 cm²