Question

if the radii of two spheres are 3cm and 4cm..
which one of the following gives the radius of the sphere whose surface area is equal to the sum of the surface areas of the given spear

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Answer 1

Given:

The radii of the two spheres are 3cm and 4cm.

To find:

Which one of the following gives the radius of the sphere whose surface area is equal to the sum of the surface areas of the given sphere.

Solution:

From given, we have the data as follows.

The radius of the first sphere is 3 cm.

Now, we will compute the surface area of the sphere whose radius is 3 cm.

Area of the sphere, A₁ = 4πr²

A₁ = 4π × 3²

A₁ = 113.1 cm²

The radius of the first sphere is 4 cm.

Now, we will compute the surface area of the sphere whose radius is 4 cm.

Area of the sphere, A₂ = 4πr²

A₂ = 4π × 4²

A₂ = 201.06 cm²

The sum of spheres is,

A₁ + A₂ = 113.1 + 201.06 = 314.16 cm²

Now, we will compute the area of the sphere whose surface area is equal to the surface area of the sum of the surface areas of the given spheres.

So, we have,

Area of the sphere, A₃ = 4πr²

A₃ = 4π × 5²

A₃ = 314.16 cm²

Therefore, the sphere whose surface area is equal to the sum of the surface areas of the given spheres is the sphere having the radius 5 cm.

Answer 2

SOLUTION

GIVEN

The radii of two spheres are 3cm and 4cm

TO DETERMINE

The radius of the sphere whose surface area is equal to the sum of the surface areas of the given spheres

FORMULA TO BE IMPLEMENTED

If r unit is the radius of a sphere then surface of the sphere is

$\sf{ = 4\pi {r}^{2} \: \: \: sq .unit}$

EVALUATION

Here the the radii of two spheres are 3cm and 4cm

So the surface area of the first sphere

$= \sf{4\pi \times {(3)}^{2} \: \: \: \: sq.cm }$

Again the surface area of second sphere

$= \sf{4\pi \times {(4)}^{2} \: \: \: \: sq.cm }$

Let the radius of the new sphere = r cm

So the surface area of the new sphere

$= \sf{4\pi \times {(r)}^{2} \: \: \: \: sq.cm }$

Hence by the given condition

$\sf{4\pi \times {(r)}^{2} = 4\pi \times {(3)}^{2} + 4\pi \times {(4)}^{2} }$

$\sf{ \implies \: {r}^{2} = {3}^{2} + {4}^{2} }$

$\sf{ \implies \: {r}^{2} = 16 + 9 }$

$\sf{ \implies \: {r}^{2} = 25}$

$\sf{ \implies \: {r}^{2} = {5}^{2} }$

$\sf{ \implies \: r = 5 }$

FINAL ANSWER

Hence the required radius = 5 cm

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