Question

Solve question 30 fast

Answer 1

☺️☺️☺️
________________________________________

$\underline{\huge{\textbf{Solution}}}$:

Given,

For a conical bucket,

Height of bucket $h= 32\: cm$

Radius of lower base of bucket $r= 16\: cm$

Radius of upper base of bucket $R = 40\: cm$

Capacity of the bucket = ?

Total Surface Area of the Bucket = ?

$\underline{\large{\textbf{Step-1}}}$:
Find the Capacity (Volume) of the Bucket

Shape of the Bucket = Shape of the frustum of a Cone

Volume of the bucket = Volume of the frustum of the cone

$Volume of the bucket\: V = \frac{1}{3}\, \pi h \, [R^{2}+r^{2}+Rr]$

$\implies V = \frac{1}{3}\times \frac{22}{7} \times 32 \times [40^{2}+16^{2}+(40 \times 16]$

$\implies V = \frac{1}{3}\times \frac{22}{7} \times 32 \times [1600+256+(640)]$

$\implies V = \frac{1}{3}\times \frac{22}{7} \times 32 \times[2496]$

$\implies V = \frac{704}{21} \times [2496]$

$\implies V = 83675\: cm^{3}$

Therefore,
Capacity of the bucket = $83675\: cm^{3}$

$\underline{\large{\textbf{Step-2}}}$:
Find the slant height of frustum of the conical bucket (s)

$Slant\: height\: s = \sqrt{(R-r)^{2}+h^{2}}$

$\implies s = \sqrt{(40-16)^{2}+32^{2}}$

$\implies s = \sqrt{(24)^{2}+32^{2}}$

$\implies s = \sqrt{(576+1024}$

$\implies s = \sqrt{(576+1024}$

$\implies s = \sqrt{1600}$

$\implies s = 40\: cm$

Slant height s = 40 cm

$\underline{\large{\textbf{Step-3}}}$:
Find the Total Surface Area of the bucket.

Total Surface Area of Bucket = Total Surface Area of Frustum of Cone

$Total\: surface\: Area = \pi[R^{2}+r^{2}+(R+r)l]$

$\implies TSA= \pi[40^{2}+16^{2}+(40+16)40]$

$\implies TSA= \pi[1600+256+(56×40)]$

$\implies TSA = \pi(1856+2240)$

$\implies TSA = \pi(4096)$

$\implies TSA = \frac{22}{7}(4096)$

$\implies TSA = 12873\: cm^{3}$

Therefore,
Total Surface Area of the Bucket = $12873\: cm^{3}$

________________________________________
✌