Question

A solid oblique cone with a slant length of 17 units is placed inside an empty cylinder with a congruent base of radius 8 units and a height of 15 units. A solid oblique cone with a slant length of 17 units is inside an empty cylinder with a congruent base of radius 8 units and a height of 15 units. What is the unfilled volume inside the cylinder? 320π cubic units 597π cubic units 640π cubic units 725π cubic units asap pls

Answer 1

$\sf\blue{Correct \ Question}$

$\sf{A \ solid \ oblique \ cone \ with \ a \ slant \ height}$

$\sf{of \ 17 \ units \ is \ placed \ inside \ an \ empty}$

$\sf{cylinder \ with \ a \ congruent \ base \ of \ radius \ 8 \ units}$

$\sf{and \ a \ height \ of \ 15 \ units. \ What \ is \ the}$

$\sf{unfilled \ volume \ inside \ the \ cylinder?}$

____________________________________

$\sf\red{\underline{\underline{Answer:}}}$

$\sf{The \ unfilled \ volume \ of \ cylinder \ is}$

$\sf{640\pi \ cubic \ units}$

$\sf\orange{Given:}$

$\sf{For \ solid \ cone,}$

$\sf{\implies{Slant \ height (l)=17 \ units}}$

$\sf{\implies{Radius (r)=8 \ units}}$

$\sf{For \ empty \ cylinder,}$

$\sf{\implies{Height (H)=15 \ units.}}$

$\sf{\implies{Radius (r)=8 \ units}}$

$\sf\pink{To \ find:}$

$\sf{The \ unfilled \ volume \ inside \ the \ cylinder.}$

$\sf\green{\underline{\underline{Solution:}}}$

$\sf{For \ cone,}$

$\boxed{\sf{Slant \ height=\sqrt{Radius^{2}+Height^{2}}}}$

$\sf{On \ squaring \ both \ sides}$

$\sf{17^{2}=8^{2}+h^{2}}$

$\sf{h^{2}=289-64}$

$\sf{h^{2}=225}$

$\sf{On \ taking \ square \ root \ of \ both \ sides.}$

$\sf{\therefore{h=15 \ units}}$

$\boxed{\sf{Volume \ of \ cone=\frac{1}{3}\times\pi\times \ r^{2}\times \ h}}$

$\sf{\therefore{Volume \ of \ cone=\frac{1}{3}\times\pi\times \ 8^{2}\times \ 15}}$

$\sf{=64\times5\times\pi}$

$\sf{=320\pi \ cubic \ units...(1)}$

$\sf{For \ cylinder,}$

$\boxed{\sf{Volume \ of \ cylinder=\pi\times \ r^{2}\times \ h}}$

$\sf{\therefore{Volume \ of \ cylinder=64\times15\times\pi}}$

$\sf{=960\pi \ cubic \ units...(2)}$

$\sf{The \ unfilled \ volume \ inside \ the \ cylinder}$

$\sf{=Volume \ of \ cylinder \ - \ Volume \ of \ cone}$

$\sf{...from \ (1) \ and \ (2)}$

$\sf{=960\pi \ - \ 320\pi}$

$\sf{=640\pi}$

$\sf\purple{\tt{\therefore{The \ unfilled \ volume \ of \ cylinder \ is}}}$

$\sf\purple{\tt{640\pi \ cubic \ units}}$

Answer 2

$\large\bf\underline{Correct \:Question:-}$

A solid oblique cone with a slant length of 17 units is inside an empty cylinder with a congruent base of radius 8 units and a height of 15 units. What is the unfilled volume inside the cylinder?

Options:-

(i) 320π cubic units

(ii) 597π cubic units

(iii) 640π cubic units

(iv)725π cubic units

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

• slent height of cone = 17 units.
• base radius of cylinder = 8 units
• height of cylinder = 15 units.

$\large\bf\underline {To \: find:-}$

• unfilled volume inside cylinder.

$\huge\bf\underline{Solution:-}$

We know that,

• slent height = 17 unit's

So, we need to find the height of cone.

Slent height (l) = √r²+ h²

So,

• l² = r² + h²

Radius of cylinder = radius of cone

radius of cone = 8units

➝ 17² = 8² + height²

➝ 238 = 64 + height²

➝ 238 - 64 = height²

➝ √225 = height

➝ height = 15

So, we get the height = 15 units

Now,

$\boxed{ \bf{volume \: of \: cone \: = \frac{1}{3} (\pi \: r {}^{2})h }}$

$\longmapsto \rm \:volume \: of \: cone \: = \frac{1}{3} ( \pi \times {64}) \times 15 \\ \\\longmapsto \rm \:volume \: of \: cone \: = \cancel \dfrac{15}{3} \pi \times 64 \\ \\ \longmapsto \rm \:volume \: of \: cone \: = 5 \times 64 \times \pi \\ \\ \longmapsto \rm \:volume \: of \: cone \: = 320 \pi \: cubic \: units$

Now,

$\boxed{ \bf{volume \: of \: cylinder = \pi \: r {}^{2}h }}$

• r = 8
• h = 15

$\longmapsto \rm \:volume \: of \: cylinder \: = \pi \times 64 \times 15 \\ \\ \longmapsto \rm \:volume \: of \: cylinder \: = 960 \pi$

Unfilled volume inside the cylinder = Volume of cylinder - volume of cone

➝ 960π - 320π

➝ 640π

So, Unfilled volume of cylinder = 640π

Option (iii) is correct

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