Question

the radius of a circle is greater than theradius of other circle by 3m.the sumof their areas is 89rm². Find the
radius of each circle


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Answer 1

ANSWER:

Given:

• Radius of one circle is 3m greater than the other circle.
• Sum of their areas is 89πm².

To Find:

• Radius of each circle.

Solution:

Let the radius of 1st circle be x.

So,

⇒ Radius of other circle = x + 3

We know that,

⇒ Area of a circle = πr²

So,

⇒ Area of 1st circle = π(x)²

And,

⇒ Area of 2nd circle = π(x + 3)²

We are given that,

⇒ Area of 1st circle + Area of 2nd circle = 89πm²

So,

⇒ π(x)² + π(x + 3)² = 89πm²

Taking π common,

⇒ π[(x²) + (x + 3)²] = 89πm²

Cancelling π on both sides,

⇒ (x²) + (x + 3)² = 89m²

We know that,

⇒ (a + b)² = a² + 2ab + b²

So,

⇒ x² + x² + 6x + 9 = 89

⇒ 2x² + 6x + 9 = 89

Transposing RHS to LHS,

⇒ 2x² + 6x + 9 - 89 = 0

⇒ 2x² + 6x - 80 = 0

Taking 2 common,

⇒ 2(x² + 3x - 40) = 0

⇒ x² + 3x - 40 = 0

On splitting the middle term,

⇒ x² + 8x - 5x - 40 = 0

⇒ x(x + 8) - 5(x + 8) = 0

⇒ (x + 8)(x - 5) = 0

⇒ x = -8 or 5

But radius can not be negative, so x = -8 gets rejected.

Hence,

⇒ x = 5

⇒ Radius of 1st circle = x = 5m

And,

⇒ Radius of 2nd circle = x + 3 = 5 + 3 = 8m

Therefore,

The radius of each circle is 5m and 8m respectively.

Formula used:

• Area of a circle = πr²
• (a + b)² = a² + 2ab + b²

Verification:

1) Radius of one circle is 3m greater than the other

⇒ 8m - 5m = 3m --------(i)

2) Sum of their areas is 89πm²

⇒ π(8)² + π(5)² ⇒ 64π + 25π ⇒ 89π --------(ii)

From (i) & (ii),

Hence Verified!!!