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The diameter of a circle is 5(2x+28) and a radius of an equal circle is 6(x+5)+x. Find the length of the diameter and the radius.

Nov 16, 2018

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Solve for x:
2 (x + 6 (x + 5)) = 5 (2 x + 28)

6 (x + 5) = 6 x + 30:
2 (6 x + 30 + x) = 5 (2 x + 28)

Grouping like terms, 6 x + x + 30 = (x + 6 x) + 30:
2 ((x + 6 x) + 30) = 5 (2 x + 28)

x + 6 x = 7 x:
2 (7 x + 30) = 5 (2 x + 28)

Expand out terms of the left hand side:
14 x + 60 = 5 (2 x + 28)

Expand out terms of the right hand side:
14 x + 60 = 10 x + 140

Subtract 10 x from both sides:
(14 x - 10 x) + 60 = (10 x - 10 x) + 140

14 x - 10 x = 4 x:
4 x + 60 = (10 x - 10 x) + 140

10 x - 10 x = 0:
4 x + 60 = 140

Subtract 60 from both sides:
4 x + (60 - 60) = 140 - 60

60 - 60 = 0:
4 x = 140 - 60

140 - 60 = 80:
4 x = 80

Divide both sides of 4 x = 80 by 4:
(4 x)/4 = 80/4

4/4 = 1:
x = 80/4

The gcd of 80 and 4 is 4, so 80/4 = (4×20)/(4×1) = 4/4×20 = 20:

x = 20    Diameter = 340  and  Radius = 170

Nov 16, 2018
#2
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If the radius of an equal circle is 6(x + 5) + x.....its diameter is twice this

So.....equating diameters, we have

5(2x + 28)   =   2 [ 6(x + 5) + x )      simplify

10x + 140  =   2 [ 6x + 30 + x ]

10x + 140  =  2 [ 7x + 30 ]

10x + 140  = 14x + 60       subtract 60, 10x from both sides

4x = 80     divide both sides by 4

x = 20

So....the diameter is  5(2*20 + 28)   =  5 (68)  =  340

And the radius is 1/2 of this  =  170   Nov 16, 2018
edited by CPhill  Nov 16, 2018