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Short answer: 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.
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.