Question

If the radii of the conical frustum bucket are 14 cm and 7 cm. If its height is 30 cm, then find its (i) total surface area and (ii) capacity of the bucket.

Answer 1

Solution :

$\underline{\bf{Given\::}}$

If the radii of the conical frustum bucket are 14 cm & 7 cm. If it's height is 30 cm.

$\underline{\bf{Explanation\::}}$

Firstly, attachment a figure of frustum bucket according to the given question,where as;

• R = 14 cm
• r = 7 cm
• h = 30 cm

Now;

Using formula of the slant height;

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

$\mapsto\sf{l=\sqrt{(14-7)^{2} + (30)^{2}}} \\\\\mapsto\sf{l=\sqrt{7^{2} + 30^{2}}} \\\\\mapsto\sf{l=\sqrt{49 + 900}} \\\\\mapsto\sf{l= \sqrt{949} }\\\\\mapsto\bf{l=30.8\:cm}$

Using formula of the total surface area of conical frustum bucket;

$\boxed{\bf{T.S.A \:of\:conical\:bucket =\pi l(R+r)+\pi R^{2} + \pi r^{2} }}}}$

$\mapsto\sf{T.S.A \:of\:bucket = \pi \times 30.8 (14 + 7) + \pi (14)^{2} + \pi (7)^{2} }\\\\\mapsto\sf{T.S.A \:of\:bucket = 3.14 \times 30.8 \times 21 + 3.14 \times 196 + 3.14 \times 49 } \\\\\mapsto\sf{T.S.A \:of\:bucket = 2030.95 + 615.44 + 153.86}\\\\\mapsto\bf{T.S.A \:of\:bucket = 2800.25\:cm^{2} }$

Now;

Using formula of the volume of the conical frustum bucket;

$\boxed{\bf{Volume\:of\:frustum\:bucket=\frac{\pi }{3} (R^{2} + r^{2} + R+r)h}}}}$

$\mapsto\sf{Volume\:_{(bucket)} =\dfrac{3.14 }{3} \bigg((14)^{2} + (7)^{2} + 14+7\bigg)}\\\\\\\mapsto\sf{Volume\:_{(bucket)} =\dfrac{3.14 }{3} \bigg(196 + 49 + 14+7\bigg)}\\\\\\\mapsto\sf{Volume\:_{(bucket)} =\dfrac{3.14 }{3} \bigg(266\bigg)}\\\\\\\mapsto\sf{Volume\:_{(bucket)} =\dfrac{3.14 }{\cancel{3}} \times \cancel{266}}\\\\\\\mapsto\sf{Volume\:_{(bucket)} = 3.14 \times 88.66}\\\\\mapsto\bf{Volume\:_{(bucket)} = 278.41\:cm^{3}}$