# A vessel is in the form of an inverted cone. Its height is 8cm and the radius of its top, which is open is 5cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5cm are dropped into the vessel, $\dfrac{1}{4}$ of the water flows out. Find the number of lead shots dropped in the vessel.

Last updated date: 19th Mar 2023

•

Total views: 307.8k

•

Views today: 3.88k

Answer

Verified

307.8k+ views

Hint: - Volume of cone $ = \dfrac{1}{3}\pi {\left( r \right)^2}h$

Given:

Height$\left( h \right)$of conical vessel$ = 8cm$

Radius$\left( r \right)$ of conical vessel$ = 5cm$

Radius$\left( {{r_1}} \right)$of the lead shots$ = 0.5cm$

Let$x$number of lead shots were dropped in the vessel

Water spilled$ = \dfrac{1}{4}$times of the volume of cone$ = x \times $volume of spherical balls

As we know volume of cone is$ = \dfrac{1}{3}\pi {\left( r \right)^2}h$

And volume of spherical balls$ = \dfrac{4}{3}\pi r_1^3$

\[

\Rightarrow \dfrac{1}{4} \times \dfrac{1}{3} \times \pi {\left( r \right)^2}h = x \times \dfrac{4}{3}\pi r_1^3 \\

\Rightarrow \dfrac{1}{4} \times \dfrac{1}{3} \times \dfrac{{22}}{7}{\left( 5 \right)^2} \times 8 = x \times \dfrac{4}{3} \times \dfrac{{22}}{7}{\left( {0.5} \right)^3} \\

\Rightarrow \dfrac{{200}}{4} = 4x \times .125 \\

\Rightarrow x = \dfrac{{200}}{{16 \times .125}} = \dfrac{{200}}{2} = 100 \\

\]

So, the number of lead shots dropped into the vessel is equal to 100.

So, 100 lead shots is the required answer.

Note: - In such types of questions the key concept we have to remember is that always remember the formula of cone and sphere which is stated above, then simplify it according to given condition we will get the number of required lead shots which is dropped into the vessel.

Given:

Height$\left( h \right)$of conical vessel$ = 8cm$

Radius$\left( r \right)$ of conical vessel$ = 5cm$

Radius$\left( {{r_1}} \right)$of the lead shots$ = 0.5cm$

Let$x$number of lead shots were dropped in the vessel

Water spilled$ = \dfrac{1}{4}$times of the volume of cone$ = x \times $volume of spherical balls

As we know volume of cone is$ = \dfrac{1}{3}\pi {\left( r \right)^2}h$

And volume of spherical balls$ = \dfrac{4}{3}\pi r_1^3$

\[

\Rightarrow \dfrac{1}{4} \times \dfrac{1}{3} \times \pi {\left( r \right)^2}h = x \times \dfrac{4}{3}\pi r_1^3 \\

\Rightarrow \dfrac{1}{4} \times \dfrac{1}{3} \times \dfrac{{22}}{7}{\left( 5 \right)^2} \times 8 = x \times \dfrac{4}{3} \times \dfrac{{22}}{7}{\left( {0.5} \right)^3} \\

\Rightarrow \dfrac{{200}}{4} = 4x \times .125 \\

\Rightarrow x = \dfrac{{200}}{{16 \times .125}} = \dfrac{{200}}{2} = 100 \\

\]

So, the number of lead shots dropped into the vessel is equal to 100.

So, 100 lead shots is the required answer.

Note: - In such types of questions the key concept we have to remember is that always remember the formula of cone and sphere which is stated above, then simplify it according to given condition we will get the number of required lead shots which is dropped into the vessel.

Recently Updated Pages

If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE