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Three concentric circles are shown. The two largest circles have radii of $12$ and $13.$ If the area of the ring between the two largest circles equals the area of the smallest circle, determine the radius of the smallest circle.

If one draws this, it may be seen that the area of the small circle plus the area of the two rings equals the area of the large circle.

The area of the large circle is 144 pi cm^2, from pi*r^2

The area of each of the rings is equal to that of the small circle, so the area of all three (circle and two rings) is 144pi cm^2, and since the three are equal, the area of the small circle is 48 pi cm^2

That means 48 pi cm^2=pi* r^2

so r^2=48

r= sqrt (48) or 4 sqrt (3) cm or about 6.9 cm

Step-by-step explanation:

More information at https://www.algebra.com/algebra/homework/Surface-area/Surface-area.faq.question.1133859.html

Hope this helped!

- 2 + 10x³ - 21x² + 19x - 4

step-by-step explanation:

each term in the second factor is multiplied by each term in the first factor

x²(- 2x² + 4x - 1) - 3x(- 2x² + 4x - 1) + 4(- 2x² + 4x - 1) ← distribute

= - 2 + 4x³ - x² + 6x³ - 12x² + 3x -8x² + 16x - 4

collect like terms

= - 2 + 10x³ - 21x² + 19x - 4 ← in standard form

-2

7 - 9 = -2

reversed:

-2 + 9 = 7