Question

if the surface area of a sphere increases by a factor of 3, by what factor does the radius change of the sphere change?

Answer 1

Answer:

√3

Explanation:

Let the radius changes to xR.

At initial, surface area = 4πR

When, surface area = 3 * (4πR)

=> 4π(xR)² = 3 * 4πR²

=> (xR)² = 3 * R²

=> x² * R² = 3 * R²

=> x² = 3

=> x = √3

Hence radius changes by factor √3

Answer 2

Given:

• The surface area of sphere is increased by factor if 3

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Need to find:

• By what factor the radius changes

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Solution:

We know,

$surface \: area \: = 4\pi {r}^{2}$

Let the radius by increased by a

Now the surface area:

$= 4\pi{(ar)}^{2} - - - - - 1$

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Also given the Final surface are is:

$= 3 \times 4\pi {r}^{2} - - - - - 2$

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Equating 1 and 2 we get,

$4\pi {(ar)}^{2} = 3 \times 4\pi {r}^{2} \\ \: \implies \: {a}^{2} \times {r}^{2} = 3 \times {r}^{2} \\ \implies \: {a}^{2} = 3$

$\red{\bold{\boxed{\large{\implies a= \sqrt{3}}}}}$

Thus the radius increases by a factor of $\sqrt{3}$