I forgot my radii!

Algebra Level pending

I have 2 2 circles with different sizes. All I know is that:

  1. The sum of the areas is 58 π 58π

  2. The sum of the circumferences is 20 π 20π

If the radii of the two circles are a a and b b respectively, submit your answer as a + b a + b .

I forgot my radii! 2

20 9.5 10 12

Ethan Mandelez by Ethan Mandelez , 15, Malaysia

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3 solutions

Dwaipayan Shikari
Nov 30, 2020

Apple way 🍎🍏 \textrm{Apple way 🍎🍏} 2 π a + 2 π b = 20 π 2πa +2πb=20π a + b = 10 a +b = \boxed{10} No need for finding for a a and b b

EDIT \textrm{EDIT}

If you want to find a a and b b a 2 + b 2 = 58 a^2 +b^2 = 58
( a + b ) 2 − 2 a b = 58 (a+b)^2 -2ab = 58

4 a b = 84 4ab=84 ( a − b ) = ± ( a + b ) 2 − 4 a b ⟹ ( a − b ) = ± 4 (a-b)= ±\sqrt{(a+b)^2 -4ab} \implies{(a-b)=±4} Solving for a a and b b a = 7 o r 3 , b = 7 o r 3 \boxed{a = 7 or 3 , b=7 or 3}

1 Helpful 1 Interesting 0 Brilliant 0 Confused

Really nice, well done! Guess this is the 'apple way 🍎🍏' to do it 👍

Ethan Mandelez - 6 months, 1 week ago

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May be . 🍎🍏

Dwaipayan Shikari - 6 months, 1 week ago
Ethan Mandelez
Nov 30, 2020

There are several methods to do this, here is the solution using simultaneous equations:

1 Helpful 0 Interesting 1 Brilliant 0 Confused

Very nice write-up. Just one comment on a different way to solve the equations: once you get to a 2 + b 2 = 58 a^2+b^2=58 and a + b = 10 a+b=10 you can square the second one to get a 2 + 2 a b + b 2 = 100 a^2+2ab+b^2=100

From here, 2 a b = 100 − 58 = 42 2ab=100-58=42 , and a 2 − 2 a b + b 2 = 58 − 42 ( a − b ) 2 = 16 \begin{aligned} a^2-2ab+b^2&=58-42 \\ (a-b)^2=16 \end{aligned}

By symmetry, we can assume a > b a>b ; so a − b = 4 a-b=4

Combining this with a + b = 10 a+b=10 gives the values of a a and b b .

I just mention this because this exact technique is used in solving cubic equations, and it's a useful trick to know.

Chris Lewis - 6 months, 2 weeks ago

Thanks for your comment, what a nice way to solve this problem! And this technique is very useful to solve more complex equations too. Again, thank you for the tip

Ethan Mandelez - 6 months, 1 week ago
Richard Desper
Nov 30, 2020

"The sum of the circumferences is 20 π 20\pi "

"If the radii of the two circles are a and b respectively, submit your answer as a + b a+b "

The sum of the circumferences is 2 π ( a + b ) 2\pi(a+b) . I don't need to find the exact values of a a or b b to answer this question. I just need to take the sum you've given me and divide by 2 π 2\pi .

1 Helpful 0 Interesting 1 Brilliant 0 Confused

Nice spot! I should have just asked for the radius of the smaller circle instead for the sum of the two....but anyways this is a very good shortcut to take, well done!

Ethan Mandelez - 6 months, 1 week ago

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