question_answer

A bucket open at the top is in the form of frustum of a cone with a capacity of \[12308.8\text{ }c{{m}^{3}}\]. 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 metal sheet used in making the bucket. (use \[\pi =3.14\])

Answer:

Given, the radii of top and bottom circular ends are 20 cm and 12 cm respectively.

And, volume of frustum (bucket) \[=12308.8\text{ }c{{m}^{3}}\]

\[\Rightarrow \frac{\pi h}{3}[{{R}^{2}}+{{r}^{2}}+Rr]=12308.8\]

\[\frac{3.14\times h}{3}[400+144+240]=12308.8\]

\[\therefore \]                  Height \[(h)=\frac{12308.8\times 3}{3.14\times 784}\]

\[=\frac{36926.4}{2461.76}=15\,cm\]

Slant height of the bucket \[(l)=\sqrt{{{h}^{2}}+{{(R-r)}^{2}}}\]

\[=\sqrt{{{(15)}^{2}}+{{(20-12)}^{2}}}\]

\[=\sqrt{225+64}=\sqrt{289}\]

\[=17\,cm\]

\[\therefore \] Area of metal sheet used in making the bucket

= Curved surface area of frustum + Base area

\[=\pi l(R+r)+\pi {{r}^{2}}\]

\[=3.14\times 17\times (20+12)+3.14\times 12\times 12\]

\[=3.14\times 17\times 32+3.14\times 144\]

\[=3.14\text{ (}544+144)\]

\[=3.14\times 688\]

\[=2160.32\text{ }c{{m}^{2}}\]