Question

To keep the lawn green and cool, Sadhna uses water sprinklers which rotate in circular shape and cover a particular area.

The diagram below shows the circular areas covered by two sprinklers:

 

Two circles touch externally. The sum of their areas is 130π sq m and the distance between their centres is 14 m.

Based on the above information, answer the following questions:

1. Obtain a quadratic equation involving R and r from above.    1
2. Write a quadratic equation involving only r.    1
3. 1. Find the radius r and the corresponding area irrigated.    2
      OR
   2. Find the radius R and the corresponding area irrigated.    2

Answer 1

(i)

We have,

R + r = 14      ...[Given]

Also,

Sum of areas = 130π m²

⇒ πR² + πr² = 130π

⇒ R² + r² = 130      ...(i)

is the required quadratic equation.

(ii)

We have,

R + r = 14

R = 14 − r

Putting in eqn (i),

⇒ (14 − r)² + r² = 130

⇒ 196 + r² − 28r + r² = 130

⇒ 2r² − 28r + 66 = 0

⇒ r² − 14r + 33 = 0

Is the required quadratic equation in r only.

(iii) (a)

We have,

⇒ r² − 14r + 33 = 0

⇒ r² − 11r − 3r + 33 = 0

⇒ r(r − 11) − 3(r − 11) = 0

⇒ (r − 11)(r − 3) = 0

⇒ r = 11 (rejected)      ...[As R > r]

So, r = 3 m

Corresponding Area irrigated = πr² = 9π m².

OR (b)

We have,

⇒ r² − 14 r + 33 = 0

⇒ r² − 11r − 3r + 33 = 0

⇒ r(r − 11) − 3(r − 11) = 0

⇒ (r − 11)(r − 3) = 0

⇒ r = 11 (rejected)      ...[As R > r]

So, r = 3 m

Now,

R + r = 14

⇒ R + 3 = 14

⇒ R = 11 m

Corresponding area irrigated = πR² = 121π m².