Maths

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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.

  • Maths -

    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

  • Maths -

    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.

  • Maths -

    ð = pi

  • Maths -

    (14 - r2)^(14 - r2)^2 + r2^2 = 130
    what is this ^ symbol represents
    please explain.I cant understand it.

  • Maths -

    madam ......please explain how did u get this step.130 pi cm^2 = pi(r1^2 + r2^2)

  • Maths -

    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

  • Maths -

    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.

  • Maths -

    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

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