Answer

Verified

432.6k+ views

Hint: Find the volume of the cone and then find the radius of the sphere from the volume of cone as they are equal. Substitute the value of radius in the equation of surface area of sphere. Simplify it and find the surface area of the sphere.

Complete step-by-step answer:

We have been given the dimensions of a cone. The radius of the base of the cone is 12 cm and the radius can be represented as ‘r’.

The height of the cone can be marked as ‘h’ which is 6 cm.

We know the volume of a cone is \[{{{}^{1}/{}_{3}}^{rd}}\]of the volume of the cylinder.

Volume of cone =\[\dfrac{1}{3}\pi {{r}^{2}}h\].

It is said that the volume of the cone is equal to the volume of the sphere.

The volume of the sphere is given as\[\dfrac{4}{3}\pi {{r}^{2}}\].

Volume of sphere = Volume of cone……….. (1)

Substitute the volume of sphere and volume of cone and find the radius of the sphere, which can be taken as ‘r’.

\[\dfrac{4}{3}\pi {{r}^{3}}=\dfrac{1}{3}\pi {{r}^{2}}h\].

Substitute the value of radius r = 12 cm and height h = 6 cm in the equation.

\[\dfrac{4}{3}\pi {{r}^{3}}=\dfrac{1}{3}\pi \times {{12}^{2}}\times 6\]. [Cancel out \[\pi \]on LHS and RHS]

\[{{r}^{3}}=\dfrac{3}{4}\times \dfrac{1}{3}\times 12\times 6\times 12\].

\[{{\pi }^{3}}=\dfrac{12\times 12\times 6}{4}=\dfrac{864}{4}=216\].

Take cube root on both LHS and RHS.

\[{{r}^{3}}=216\].

\[r=\sqrt[3]{216}=\sqrt[3]{6\times 6\times 6}=6cm\].

Therefore, radius of the sphere, r = 6 cm.

Now what we need to find is the surface area of the sphere.

The surface area of a sphere is given by\[4\pi {{r}^{2}}\], where r = 6 cm.

Surface area of sphere =\[4\pi {{r}^{2}}\].

=\[4\times \dfrac{22}{7}\times 6\times 6\] [\[\pi \]can be taken as\[{}^{22}/{}_{7}\]]

= 452.57 sq. cm.

= 452.57\[c{{m}^{2}}\].

The surface area of sphere = 452.57\[c{{m}^{2}}\].

Note: A sphere is perfectly symmetrical. All the points on the surface are the same distance ‘r’ from the center. The main difference between a sphere and circle is that spheres represent 3 dimensionally, which circles are mentioned in 2 dimensions. But radius is taken the same way.

Complete step-by-step answer:

We have been given the dimensions of a cone. The radius of the base of the cone is 12 cm and the radius can be represented as ‘r’.

The height of the cone can be marked as ‘h’ which is 6 cm.

We know the volume of a cone is \[{{{}^{1}/{}_{3}}^{rd}}\]of the volume of the cylinder.

Volume of cone =\[\dfrac{1}{3}\pi {{r}^{2}}h\].

It is said that the volume of the cone is equal to the volume of the sphere.

The volume of the sphere is given as\[\dfrac{4}{3}\pi {{r}^{2}}\].

Volume of sphere = Volume of cone……….. (1)

Substitute the volume of sphere and volume of cone and find the radius of the sphere, which can be taken as ‘r’.

\[\dfrac{4}{3}\pi {{r}^{3}}=\dfrac{1}{3}\pi {{r}^{2}}h\].

Substitute the value of radius r = 12 cm and height h = 6 cm in the equation.

\[\dfrac{4}{3}\pi {{r}^{3}}=\dfrac{1}{3}\pi \times {{12}^{2}}\times 6\]. [Cancel out \[\pi \]on LHS and RHS]

\[{{r}^{3}}=\dfrac{3}{4}\times \dfrac{1}{3}\times 12\times 6\times 12\].

\[{{\pi }^{3}}=\dfrac{12\times 12\times 6}{4}=\dfrac{864}{4}=216\].

Take cube root on both LHS and RHS.

\[{{r}^{3}}=216\].

\[r=\sqrt[3]{216}=\sqrt[3]{6\times 6\times 6}=6cm\].

Therefore, radius of the sphere, r = 6 cm.

Now what we need to find is the surface area of the sphere.

The surface area of a sphere is given by\[4\pi {{r}^{2}}\], where r = 6 cm.

Surface area of sphere =\[4\pi {{r}^{2}}\].

=\[4\times \dfrac{22}{7}\times 6\times 6\] [\[\pi \]can be taken as\[{}^{22}/{}_{7}\]]

= 452.57 sq. cm.

= 452.57\[c{{m}^{2}}\].

The surface area of sphere = 452.57\[c{{m}^{2}}\].

Note: A sphere is perfectly symmetrical. All the points on the surface are the same distance ‘r’ from the center. The main difference between a sphere and circle is that spheres represent 3 dimensionally, which circles are mentioned in 2 dimensions. But radius is taken the same way.

Recently Updated Pages

Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred

The branch of science which deals with nature and natural class 10 physics CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Trending doubts

What type of defect is shown by NaCl in a Stoichiometric class 12 chemistry CBSE

Difference Between Plant Cell and Animal Cell

Distinguish between tetrahedral voids and octahedral class 12 chemistry CBSE

Derive an expression for electric potential at point class 12 physics CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

What is BLO What is the full form of BLO class 8 social science CBSE

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

Give 10 examples for herbs , shrubs , climbers , creepers

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE