Answer

Verified

458.4k+ views

Hint- Area of top of the hemispherical part\[ = \pi {r^2}\] sq. Unit, Curved surface area of the hemispherical part\[ = 2\pi {r^2}\] sq. unit

Consider the diagram shown above.

It is given that a hemisphere of diameter $l$unit is cut out from the top face of the cubical wooden block.

$\therefore $Radius of hemisphere\[\left( r \right) = \dfrac{{{\text{diameter}}}}{2} = \dfrac{l}{2}\]unit.

Therefore surface area of remaining solid

$ \Rightarrow $Surface area of the cuboidal box who’s each edge is of length $l$ unit$ - $Area of top of the hemispherical part$ + $Curved surface area of the hemispherical part.

$ \Rightarrow $Surface area of the cuboidal box who’s each edge is of length $l$ unit\[ = 6{l^2}\]sq. Unit

$ \Rightarrow $Area of top of the hemispherical part\[ = \pi {r^2}\] sq. Unit

$ \Rightarrow $Curved surface area of the hemispherical part\[ = 2\pi {r^2}\] sq. Unit

Therefore surface area$\left( A \right)$ of the remaining solid\[ = 6{l^2} - \pi {r^2} + 2\pi {r^2}\]

\[ = 6{l^2} - \pi {\left( {\dfrac{l}{2}} \right)^2} + 2\pi {\left( {\dfrac{l}{2}} \right)^2} = 6{l^2} - \pi \left( {\dfrac{{{l^2}}}{4}} \right) + \pi \left( {\dfrac{{{l^2}}}{2}} \right)\]

\[A = 6{l^2} + \pi \left( {\dfrac{{{l^2}}}{4}} \right) = \dfrac{{{l^2}}}{4}\left( {\pi + 24} \right)\] Sq. Unit

So, this is the required surface area of the remaining solid.

Note- In such types of questions after cutting the required portion according to the question then the remaining portion we found is not in a standard shape so we cannot determine surface area of the remaining solid directly, so we use the concept which is written above to determine the surface area of the remaining solid, then simplify we will get the required answer.

Consider the diagram shown above.

It is given that a hemisphere of diameter $l$unit is cut out from the top face of the cubical wooden block.

$\therefore $Radius of hemisphere\[\left( r \right) = \dfrac{{{\text{diameter}}}}{2} = \dfrac{l}{2}\]unit.

Therefore surface area of remaining solid

$ \Rightarrow $Surface area of the cuboidal box who’s each edge is of length $l$ unit$ - $Area of top of the hemispherical part$ + $Curved surface area of the hemispherical part.

$ \Rightarrow $Surface area of the cuboidal box who’s each edge is of length $l$ unit\[ = 6{l^2}\]sq. Unit

$ \Rightarrow $Area of top of the hemispherical part\[ = \pi {r^2}\] sq. Unit

$ \Rightarrow $Curved surface area of the hemispherical part\[ = 2\pi {r^2}\] sq. Unit

Therefore surface area$\left( A \right)$ of the remaining solid\[ = 6{l^2} - \pi {r^2} + 2\pi {r^2}\]

\[ = 6{l^2} - \pi {\left( {\dfrac{l}{2}} \right)^2} + 2\pi {\left( {\dfrac{l}{2}} \right)^2} = 6{l^2} - \pi \left( {\dfrac{{{l^2}}}{4}} \right) + \pi \left( {\dfrac{{{l^2}}}{2}} \right)\]

\[A = 6{l^2} + \pi \left( {\dfrac{{{l^2}}}{4}} \right) = \dfrac{{{l^2}}}{4}\left( {\pi + 24} \right)\] Sq. Unit

So, this is the required surface area of the remaining solid.

Note- In such types of questions after cutting the required portion according to the question then the remaining portion we found is not in a standard shape so we cannot determine surface area of the remaining solid directly, so we use the concept which is written above to determine the surface area of the remaining solid, then simplify we will get the required answer.

Recently Updated Pages

What number is 20 of 400 class 8 maths CBSE

Which one of the following numbers is completely divisible class 8 maths CBSE

What number is 78 of 50 A 32 B 35 C 36 D 39 E 41 class 8 maths CBSE

How many integers are there between 10 and 2 and how class 8 maths CBSE

The 3 is what percent of 12 class 8 maths CBSE

Find the circumference of the circle having radius class 8 maths CBSE

Trending doubts

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Which are the Top 10 Largest Countries of the World?

In 1946 the Interim Government was formed under a Sardar class 11 sst CBSE

10 examples of law on inertia in our daily life

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

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

Change the following sentences into negative and interrogative class 10 english CBSE