Really hard questions...
1. XY and YZ are diameters of two tangent circles and XY = 10 cm. Find YZ if the area of hte shaded region equals the area of hte unshaded region.
The answer is: 24.14, but how and why?
2. Find the diameters of the two smaller circles if the shaded area is 4/9 of the area of the larger circle which has a diameter of 12 cm.
Notes: These drawings are not EXACT as the real one because I used paint, however they really close.
The answer is 4 cm and 8 cm, but how and why...?
These questions are really hard. Nobody in my class were able to answer them.
ok.. here we go...
let's say the dist YZ is x cm, then the diameter of the larger circle is (10+x) cm. And the area is pi [(10+x)/2]^2. Similarly, the area of the smaller circle can be found ie pi(x/2)^2. We can then find an expression for the shaded area and equate that to the area of the smaller circle. You should get a wuadratic equation which can be reduced to
x^2 - 20x -100 = 0.
Solving this would give you the answer as required.
As for the second question, I think you should work on it alonmg similar lines as per question 1.
Hope it helps. Let me know if u need further clarifications.
Originally Posted by AlphaRock
I've modified your sketch a little bit.
1. Let x denote the length of the diameter of the upper circle then (12-x) is the length of the diameter of the lower circle.
2. The shaded region is the difference of the large circle and the two small circles:
3. Expand the brackets, collect like terms. You'll get a quadratic equation:
Solve for x.