I am very greatefull to you if you guide me to solve this problem.
Two circles touch externally. The sum of their areas is 130π cm2 and the distance between their centers is 14 cm. find the radii of the circles?
I tried to solve this problem like this:
Area of the circle = πr2
130πcm2 = 22/7xrxr
130x22/7 =22/7xrxr
22/7xr2 =130x22/7
r2 =130x22x7/7
r2 = 130x22x7/7x22
r2 =130
r =√130
r=√2x5x13.
I tried to do this in the above way. But I couldn't get the answer further.
Call the radii r1 and r2.
The center-to-center distance is
r1 + r2 = 14
The combined area is
130 pi cm^2 = pi(r1^2 + r2^2);
therefore
130 = r1^2 + r2^2
The two equations for r1 and r2 can easily be solved by substitution.
(14 - r2)^2 + r2^2 = 130
196 -28r2 + 2r2^2 = 130
r2^2 -14r2 + 33 = 0
(r2 - 11)(r2 -3) = 0
The radii are 3 and 11 cm
r1= first radius
r2= second radius
r1+r2=14
r2=14-r1
The sum of circles areas is:
r1^2 * ð + r2^2 * ð = 130ð
ð * ( r1^2 + r2^2 ) = 130ð Divide both sides with ð
( r1^2 + r2^2 ) = 130
r1^2 + ( 14 - r1 )^2 = 130
r1^2 + 14^2 - 2 * r1 *14 + r1^2 = 130
r1^2 + 196 - 28 * r1 + r1^2 = 130
2 * r1^2 - 28 * r1 + 196 - 130 = 0
2 * r1^2 - 28 * r1 + 66 = 0 Divide both sides with 2
r1^2 - 14 * r1 + 33 = 0
This equation have 2 solutions:
r1 = 3
and
r1 = 11
When r1 = 3
r2 = 14 - r1 = 14 - 3 = 11
When r1 = 11
r2 = 14 - r1 = 14 - 11 = 3
So radii is :
3 cm and 11 cm
The sum of circles areas:
3^2 * ð + 11^2 * ð =
9 *ð + 121 * ð = 130ð cm^2
P.S.
If you don't know solve equation:
r1^2 - 14 * r1 + 33 = 0
In google type:
Quadratic equation online
When you see list of results click on:
Free Online Quadratic Equation Solver: Solve by Quadratic Formula
When page be open in rectangle type:
r1^2 - 14 * r1 + 33 = 0
and click option solve it!
You will see solution step-by-step.
ð = pi
(14 - r2)^(14 - r2)^2 + r2^2 = 130
what is this ^ symbol represents
please explain.I cant understand it.
madam ......please explain how did u get this step.130 pi cm^2 = pi(r1^2 + r2^2)
how did u get this step.130 pi cm^2 = pi(r1^2 + r2^2)
========================
^ means "to the power"
so
r^2 = radius squared
pi r^2 = pie times radius squared
= area of circle of radius r
similar to drwls ......
radius of smaller -- r
radius of larger --- 14-r
πr^2 + π(14-r)^2 = 130π
r^2 + 196 - 28r + r^2 = 130
2r^2 - 28r + 66 = 0
r^2 - 14r + 33 = 0
(r-3)(r-11) = 0
r = 3 or r = 11
sub back in my definitions.
Radius of 1 circle = x,radius of2 circle=14-x. Now,area of circle = piR2. So,pi(x2)+pi(14-x)2=130pi. Pi(x2+196-28x+x2)=130pi. Pi at both sides is cancelled So 2x2-28x+196-130=0. 2(x2-14x+33)=0. X2-11x-3x+33=0. X(x-11)-3(x-11)=0. So, x= 3,11
You're on the right track, but let's continue solving the problem.
You've correctly obtained the equation: r^2 = 130.
To find the radius, we need to take the square root of both sides of the equation:
√r^2 = √130
The square root of r^2 is simply r, so we have:
r = √130
Now, let's simplify the value of √130:
r ≈ 11.40
Therefore, the radius of each circle is approximately 11.40 cm.