Question

$\huge\mathcal{Question}$

A cylindrical bucket of height 32 cm and base radius 18 cm is filled with sand .This bucket is emptied on the ground and a conical heap of sand is formed . If the height of the conical heap is 24cm , Find the radius and slant height of the heap .

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Answer 1

Step-by-step explanation:

Height of cylindrical bucket(h¹)=32 cm

Radius of the base of the bucket (r¹)=18 cm

∴Volume of the sand in the cylindrical bucket=πr²1h1

Height of conical heap (h²)=24 cm

let the radius of the conical heap=r²

∴Volume of the sand in conical heap=1/3πr²2h2

According to the question

The volume of the sand in the cylindrical bucket=Volume of the sand in the conical shape

πr²1h1 = 1/3πr²2h2

⇒π×(18)²×32=⅓π×r²2×24

⇒r²2 = 3×18²×32

⇒r²2 =18²×4

⇒r² =18×2=36cm

Slant height of heap= √r²2 + h²2

⇒√36²+24²

⇒√1296+576

⇒√1872

⇒ √144×13

⇒12√13cm.

Answer 2

GIVEN :-

• Height of the cylindrical bucket = 32 cm

• Base radius of the cylindrical bucket = 18 cm

• Height of the conical heap = 24 cm

TO FIND :-

• Radius of the conical heap .

• Slant height of the conical heap .

SOLUTION :-

   Formula to find the volume of cylinder = $\sf\pi r^2h$

• Volume of the sand in cylindrical bucket ,

                $\longrightarrow\sf \pi \times 18\times 18 \times 32\\\\\longrightarrow 10368 \pi \ cm^3$

    Formula to find the volume of cone = $\sf\dfrac{1}{3}\pi r^2h$

• Volume of sand in the conical heap ,

                     

                     $\longrightarrow\sf \dfrac{1}{3}\pi r^2 \times 24 \\\\\longrightarrow 8\pi r^2$

   According to the question ,

    Volume of the sand in cylindrical bucket = Volume of the  sand

                                                                            in conical heap

           

                 $\longrightarrow\sf 10368\pi =8 \pi r^2 \\\\\longrightarrow 10368 = 8r^2 \\\\\longrightarrow r^2 = 10368\div8\\\\\longrightarrow r^2 = 1296 \\\\\longrightarrow r = \sqrt{1296} \\\\\longrightarrow \bf r = 36$

   Radius of the conical heap = 36 cm

   We know that ,

           $\sf l^2 = h^2+r^2$

     $\longrightarrow\sf l^2 = 24^2+36^2 \\\\\longrightarrow l =\sqrt{576+1296} \\\\\longrightarrow l = \sqrt{1872 } \\\\\longrightarrow l =43.267$

  Slant height of the conical heap = 43.267 cm