Asap a circle has a radius of 10.9 cm. if the area is multiplied by 6, what happens to the radius?

Short Radius of new circle = sqrt(6) * the original radius
 
Area = pi * r^2
A1 = 6*Area = pi * (k * r) ^2
A1 = 6*Area = k^2 * pi * r^2 Set up a proportion.

Proportion


Comment
The As Cancel, the pis cancel the r^2 cancel.

1/6 = 1 / k^2  Cross multiply
k^2 = 6 Take the square root of both sides.
sqrt(k^2) = sqrt(6)
k = sqrt(6)

Conclusion
The radius increases by a factor of sqrt(6) times.

Discussion. You may not like this method very much, but you will do a lot of it in Physics or Chemistry. You may prefer the second method.

Step One
Find the area of the original circle
Area = pi * r^2
r = 10.9
pi = 3.14
Area = pi * r^2
Area = 3.14 * 10.9^2
Area = 3.14 * 118.81
Area = 373.1

Step 2
Multiply this area by 6
Area1 = 6 * Area
Area1 = 6 * 373.1
Area1 = 2238.38

Step 3
Find the radius of the bigger circle
Area1 = 2238.38
pi = 3.14
r = ????

Area1 = pi * r^2
2238.38 = 3.14 * r^2 Divide both sides by 3.14
2238/3.14 = r^2
r^2 = 712.86              Take the square root of both sides.
sqrt(r^2) = sqrt(712.86)
r = 26.6994

Step 4
Divide by the original r
r/10.9 = 26.6994 / 10.9 = 2.44948

Here's where this get's a little messy. How do you know what to do with this. Perhaps just saying that the new radius = the original radius * 2.44958

But try taking the square root of 6 to see what happens.
sqrt(6) = 2.449489 is what you get, so the radius increases by sqrt(6) in size.

Comment
No matter which way you do this, it looks like a messy problem just because square root 6 is not easily recognized.