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Thread: Quick maths

20040912, 16:12 #1
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Quick maths
1/ The difference of 2 numbers is 8 & their product is 13. What is the sum of their squares?
2/ In the image below the shaded rectangle is 3cm x 6cm. What is the radius of the circle?

20040912, 19:13 #2
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Re: Quick maths
1) <span style="backgroundcolor: #FFFF00; color: #FFFF00; fontweight: bold">90, combine, solve quadratic</span hide>
Steve

20040912, 19:24 #3
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Re: Quick maths
2) <span style="backgroundcolor: #FFFF00; color: #FFFF00; fontweight: bold">the radius is 15 based on the way the picture is drawn. A radius of 3 is also possible, but it would not look like the picture</span hide>
Steve

20040913, 01:57 #4
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Re: Quick maths
1) Without having to do any quadratic manipulation:
<span style="backgroundcolor: #FFFF00; color: #FFFF00; fontweight: bold">a

20040913, 09:02 #5
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Re: Quick maths
<img src=/S/clapping.gif border=0 alt=clapping width=19 height=23>
Well done Steve both correct.

20040913, 09:47 #6
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Re: Quick maths
OK Tony  that's a gauntlet! <img src=/S/grin.gif border=0 alt=grin width=15 height=15> If Steve's right on 2) then I must be wrong, and as the poser of the puzzle the onus is upon thee to cough up with the proof. I humbly await your immediate reply.
Alan

20040913, 10:24 #7
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Re: Quick maths
<span style="backgroundcolor: #FFFF00; color: #FFFF00; fontweight: bold">Draw a new rectangle inside the circle as shown in the image below.
Length of sides of the new rectangle are (r6) and (r3)
You can then use Pythagoras' Theorem to calculate the radius
(r6)

20040913, 12:01 #8
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Re: Quick maths
Thank you <img src=/S/thankyou.gif border=0 alt=thankyou width=40 height=15>. And I (reluctantly <img src=/S/grin.gif border=0 alt=grin width=15 height=15>) concede <img src=/S/bow.gif border=0 alt=bow width=15 height=15>. Now I have to go back and work out my mistake. Quite frankly, I can't see any error in my logic for solving it. Maybe it's in my maths?
Alan
Edited  Yes, I see the folly of my ways. Deforming the rectangle into a square of the same area is no guarantee that its diagonal will span between the corner of the big square and the circumference of the circle. In fact, it doesn't! For a radius of 15, such a square has an area of 19.30, or for an area of 18, the corresponding radius is 14.485.

20040913, 12:12 #9
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Re: Quick maths
The radius of 3 will not yield negative dimensions. It is a valid interpretation:
Steve

20040913, 12:28 #10
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Re: Quick maths
I agree that 3 is a valid answer if the shaded rectangle is not confined within the top left corner. The negative dimension I was referring to was for the 2nd rectangle drawn inside the circle with sides of (r6) and (r3).

20040913, 14:56 #11
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Re: Quick maths
I didn't solve it the same way you did (using figures).
I solved it using the general eqn of a circle:
(xx')