Question
The sum of the circumferences of two circles is 42 pie cm and the sum of their areas is 225 pie cm^2. Find the radius of each circle?
Answers
GPT 3.5
Let the radius of the first circle be r1 and the radius of the second circle be r2.
We know that the circumference of a circle is given by 2πr and the area of a circle is given by πr^2.
Given that the sum of the circumferences is 42π, we have:
2πr1 + 2πr2 = 42π
=> 2π(r1 + r2) = 42π
=> r1 + r2 = 21
Given that the sum of the areas is 225π, we have:
πr1^2 + πr2^2 = 225π
=> r1^2 + r2^2 = 225
Now, we can use the equations r1 + r2 = 21 and r1^2 + r2^2 = 225 to solve for the values of r1 and r2.
From r1 + r2 = 21, we can write:
r1 = 21 - r2
Substitute this into r1^2 + r2^2 = 225:
(21 - r2)^2 + r2^2 = 225
=> 441 - 42r2 + r2^2 + r2^2 = 225
=> 2r2^2 - 42r2 + 216 = 0
=> r2^2 - 21r2 + 108 = 0
=> (r2 - 9)(r2 - 12) = 0
So, r2 = 9 or r2 = 12.
If r2 = 9, then r1 = 21 - 9 = 12.
If r2 = 12, then r1 = 21 - 12 = 9.
Therefore, the radius of the first circle is 12 cm and the radius of the second circle is 9 cm.
We know that the circumference of a circle is given by 2πr and the area of a circle is given by πr^2.
Given that the sum of the circumferences is 42π, we have:
2πr1 + 2πr2 = 42π
=> 2π(r1 + r2) = 42π
=> r1 + r2 = 21
Given that the sum of the areas is 225π, we have:
πr1^2 + πr2^2 = 225π
=> r1^2 + r2^2 = 225
Now, we can use the equations r1 + r2 = 21 and r1^2 + r2^2 = 225 to solve for the values of r1 and r2.
From r1 + r2 = 21, we can write:
r1 = 21 - r2
Substitute this into r1^2 + r2^2 = 225:
(21 - r2)^2 + r2^2 = 225
=> 441 - 42r2 + r2^2 + r2^2 = 225
=> 2r2^2 - 42r2 + 216 = 0
=> r2^2 - 21r2 + 108 = 0
=> (r2 - 9)(r2 - 12) = 0
So, r2 = 9 or r2 = 12.
If r2 = 9, then r1 = 21 - 9 = 12.
If r2 = 12, then r1 = 21 - 12 = 9.
Therefore, the radius of the first circle is 12 cm and the radius of the second circle is 9 cm.