Question

A bucket open at the top is in the form of frustum of a cone with a capacity of 12308.8 cm cube.The radii of the top and bottom circular ends are 20cm and 12cm respectively. Find the height of the bucket and the area of metal sheet used in making the bucket (pie=3.14)​

Answer 1

Question :

A bucket open at the top is in the form of frustum of a cone with a capacity of 12308.8 cm³. 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 (pie=3.14)

Answer:

Height of the bucket be 15 cm and area of metal sheet used in making the bucket be 1708.16 cm³.

Step-by-step explanation:

Given:-

~ Capacity of frustum or capacity of bucket be 12308.8 cm³.

~ Radius upper portion R -20 cm

~ radius lower portion r - 12 cm.

$\tt \: volume \: of \: frustum \: = \frac{1}{3} \pi h( {R}^{2} + {r}^{2} + R \times r) = 12308.8 \: {cm}^{3}$

$\tt \: 12308.8 = \frac{ \pi h}{3}( {R}^{2} + {r}^{2} + R \times r) \\ \: \tt12308.8 = \frac{ \pi h}{3}( {20}^{2} + {12}^{2} + 20 \times 12) .$

$\: \tt h = \frac{12308.8 \times 3}{ \pi(400 + 144 + 240)} \\ \: \tt h = \frac{12308.8 \times 3}{3.14 \times 784} \\ \: \tt h = 15 \: cm .$

Height is 15 Cm.

Now,

Slant height =?

$\tt \: l = \sqrt{ {h}^{2} + {(R- r)}^{2} } \\ \: \: \: \: \tt = \sqrt{ {15}^{2} + {(20 - 12)}^{2} } . \\ \: \: \: \: \tt = \sqrt{255 + 64} \\ \: \: \: \: \tt= \sqrt{289}$

$\tt \: \: \: \: = \sqrt{17 \times 17} \\ \: \: \: \: \tt = 17 \: cm.$

Putting the value Height and slant height in it, we get,

Area of metal sheet,

$\: \: \: \: \: = \tt \pi l(r + r) \\ \: \: \: \: \: \tt= 3.14 \times 17(20 + 12) \\ \: \: \: \: \: \tt = 1708.16 \: {cm}^{3}$

Therefore, the frustum of height is 15 cm and area metal sheet used to making is 1708.16 cm³.

Some information :

$\tt\star{CSA \:of\: Frustum = \pi \:l (R+r) }$

$\tt{here \:l \:is \:slant \:height \:of\: frustum.}$

$\tt \star{l}^{2}= {h}^{2}+(R-r)^{2}$

$\tt\star\frac{1}{3} \pi h( {R}^{2} + {r}^{2} + R \times r)$