Question

if the radii of the ends of a bucket are 5cm and 15cm and it is 24 cm high then the surface area is ​

Answer 1

The surface are of bucktet= CSA of frustum

• Radius of top (R)= 15cm
• radius of bottom 2(r)= 5cm
• Height (h)= 24cm

we gave to find out the surface area of the bucket !

Surface area = πl(R+r)

where ,

$\sf\ l= \sqrt{R-r)^2+h^2}$

Now first find the slant height of frustum:-

$\implies\sf\ l= \sqrt{15-5)^2+(24)^2}\\ \\ \implies\sf\ l= \sqrt{(10)^2+(24)^2}\\ \\ \implies\sf\ l= \sqrt{100+576}\\ \\ \implies\sf\ l= \sqrt{676}\\ \\ \implies{\red{\sf\ \ l= 26cm}}$

Now surface area:-

$\implies\sf\ S.A= \dfrac{22}{7}\times (15+5)\times 26\\ \\ \\ \implies\sf\ S.A= \dfrac{22\times 20\times 26}{7}\\ \\ \\ \implies\sf\ S.A= \dfrac{440\times 26}{7}\\ \\ \\ \implies\sf\ S.A= \cancel{\dfrac{11440}{7}}\\ \\ \\ \implies{\boxed{\purple{\sf\ Surface\ area= 1634.28 cm^2}}}$

$\underline{\bigstar{\sf The\ surface\ area\ of\ bucket \ is \ 1634.28cm^2}}$

Answer 2

${\large{\bold{\rm{\underline{Given \; that}}}}}\; \; \; \red \bigstar$

✠ The radii of the ends of a bucket are 5 cm and 15cm and it is 24 cm high.

$\; \; \; \; \; \; \; \; \; \; \; \; \; \;{\sf{Means}}$

✮ Radius of the top of bucket = 15 cm

✮ Radius of the bottom of bucket = 5 cm

✮ Height of the bucket = 24 cm.

${\large{\bold{\rm{\underline{To \; find}}}}}\; \; \; \red \bigstar$

✠ Surface area of the bucket.

${\large{\bold{\rm{\underline{Solution}}}}}\; \; \; \red \bigstar$

✠ Surface area of the bucket = 1634.28 cm²

${\large{\bold{\rm{\underline{Using \; concepts}}}}}\; \; \; \red \bigstar$

✠ Formula to find surface area (cylinder)

✠ Formula to find slant height

${\large{\bold{\rm{\underline{Using \; formula}}}}}\; \; \; \red \bigstar$

✠ Surface area (cylinder) = πl(R+r)

✠ Slant height = √(R-r)² + h²

${\large{\bold{\rm{\underline{Full \; Solution}}}}}\; \; \; \red \bigstar$

~ Firstly let us find the slant height..!

➙ Slant height = √(R-r)² + h²

➙ Slant height = √(15-5)² + 24²

➙ Slant height = √(10)² + 24²

➙ Slant height = √100 + 24²

➙ Slant height = √100 + 24²

➙ Slant height = √100 + 576

➙ Slant height = √676

➙ Slant height = 26 cm

~ Now let's find surface area..!

➙ Surface area = πl(R+r)

➙ Surface area = 3.14(26)(15+5)

➙ Surface area = 3.14 × 26 (15+5)

➙ Surface area = 81.64 (15+5)

➙ Surface area = 81.64 (20)

➙ Surface area = 81.64 × 20

➙ Surface area = 1634.28 cm²

${\large{\bold{\rm{\underline{Additional \; knowledge}}}}}\; \; \; \red \bigstar$

Some formulas related to cylinder

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Volume \: of \: cylinder \: = \: \pi r^{2}h}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Surface \: area \: of \: cylinder \: = \: 2 \pi rh + 2 \pi r^{2}}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Lateral \: area \: of \: cylinder \: = \: 2 \pi rh}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Base \: area \: of \: cylinder \: = \: \pi r^{2}}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Height \: of \: cylinder \: = \: \dfrac{v}{\pi r^{2}}}}}$