# Some questions to try

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The area of the largest right angle triangle that will fit inside a circle is 18 square units. What is the area of the circle?

1!2!3!4!5!6! has how many factors?

Is the product of the cubes of the roots of the equation: x^2 + 7x + 22 = 0 a positive or negative number?

Is 9^n - 1 always divisible by 8? If it is, prove that it is, if it isn't, give a counterexample.

1!2!3!4!5!6! has how many factors?

Is the product of the cubes of the roots of the equation: x^2 + 7x + 22 = 0 a positive or negative number?

Is 9^n - 1 always divisible by 8? If it is, prove that it is, if it isn't, give a counterexample.

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#2

(Original post by

The area of the largest right angle triangle that will fit inside a circle is 18 square units. What is the area of the circle?

1!2!3!4!5!6! has how many factors?

Is the product of the cubes of the roots of the equation: x^2 + 7x + 22 = 0 a positive or negative number?

Is 9^n - 1 always divisible by 8? If it is, prove that it is, if it isn't, give a counterexample.

**mik1a**)The area of the largest right angle triangle that will fit inside a circle is 18 square units. What is the area of the circle?

1!2!3!4!5!6! has how many factors?

Is the product of the cubes of the roots of the equation: x^2 + 7x + 22 = 0 a positive or negative number?

Is 9^n - 1 always divisible by 8? If it is, prove that it is, if it isn't, give a counterexample.

The roots a,b satisfy ab = 22, so (ab)^3 = 22^3 > 0.

9^n = 1^n mod 8 = 1 mod 8, thus 9^n -1 = 0 mod 8.

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#3

(Original post by

1!2!3!4!5!6! has how many factors?

**mik1a**)1!2!3!4!5!6! has how many factors?

= 2*3*(2*2)*5*(3*2)

= 2^4 * 3^2 * 5.

So the factors of 1!2!3!4!5!6! are all the numbers of the form

2^i * 3^j * 5^k,

where 0 <= i <= 4, 0 <= j <= 2 and 0 <= k <= 1. There are 5*3*2 = 30 such numbers.

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#4

(Original post by

The area of the largest right angle triangle that will fit inside a circle is 18 square units. What is the area of the circle?

**mik1a**)The area of the largest right angle triangle that will fit inside a circle is 18 square units. What is the area of the circle?

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#5

**mik1a**)

The area of the largest right angle triangle that will fit inside a circle is 18 square units. What is the area of the circle?

1!2!3!4!5!6! has how many factors?

Is the product of the cubes of the roots of the equation: x^2 + 7x + 22 = 0 a positive or negative number?

Is 9^n - 1 always divisible by 8? If it is, prove that it is, if it isn't, give a counterexample.

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#7

(Original post by

does the middle q have any real roots?

**lgs98jonee**)does the middle q have any real roots?

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IT has complex roots which can be multiplied... as the roots are conjunctuates the imaginary parts dissapear when you find their product or sum. (neat huh)

The sum of the roots of ax^2 + bx + c = 0 is always -b/a, and the product is always c/a, neat thing to know that will help avoid a lot of quadratic equationing.

The sum of the roots of ax^2 + bx + c = 0 is always -b/a, and the product is always c/a, neat thing to know that will help avoid a lot of quadratic equationing.

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#9

If you didnt think of that, you could of course use the formula to get

(1/2[-7+sqrt(-39)])^3*(1/2[-7-sqrt(-39)])^3

Which is the difference of 2 squares giving

(88/4)^3 = 22^3.

(1/2[-7+sqrt(-39)])^3*(1/2[-7-sqrt(-39)])^3

Which is the difference of 2 squares giving

(88/4)^3 = 22^3.

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A chord of length sqrt 3 divided a circle of radius 1 into two regions. Find the area of the largest rectangle which can be inscribed in the smaller region.

I tried this one about 6 months ago and didn't do too well, couldn't do the maths when it came to finding the constraints of the rectangle. Could see the calculus at the end of the tunnel, but couldn't get there!

I tried this one about 6 months ago and didn't do too well, couldn't do the maths when it came to finding the constraints of the rectangle. Could see the calculus at the end of the tunnel, but couldn't get there!

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#11

(Original post by

A chord of length sqrt 3 divided a circle of radius 1 into two regions. Find the area of the largest rectangle which can be inscribed in the smaller region.

I tried this one about 6 months ago and didn't do too well, couldn't do the maths when it came to finding the constraints of the rectangle. Could see the calculus at the end of the tunnel, but couldn't get there!

**mik1a**)A chord of length sqrt 3 divided a circle of radius 1 into two regions. Find the area of the largest rectangle which can be inscribed in the smaller region.

I tried this one about 6 months ago and didn't do too well, couldn't do the maths when it came to finding the constraints of the rectangle. Could see the calculus at the end of the tunnel, but couldn't get there!

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#13

OK, got a new result.

From Fig 1, and the dimensions given, it can be seen that the distance from the circle centre to the chord is ½.

In Fig 2, the dist h' = ½, and the height of the rectangle in the smaller region is h and the half-width of that rectangle is b.

From Fig 2, we can see that b² + (h+h')² = r², where r is the radius of the circle and r=1.

The maths bit

b² + (h+h')² = r²

b² + (h+½)² = 1

b² = 1 - (h+½)²

==========

Area of rectangle is A,

A = 2bh

A = 2h{1 - (h+½)²}^(½)

dA/dh = 2{1 - (h+½)²}^(½) - 2h.½.2(h+½){1 - (h+½)²}^(-½) = 0

{1 - (h+½)²}^(½) = h.(h+½){1 - (h+½)²}^(-½)

{1 - (h+½)²} = h.(h+½)

1 - h² - h - ¼ = h² + h/2

2h² + 1.5h - 0.75 = 0

h = -1.5 ± rt(1.5² + 4.2.0.75) / 4

h = {-1.5 ± rt(8.25) }/4

h = -0.375 + 0.718

h = 0.3443

=======

b² = 1 - (h+½)²

b² = 1 - 0.8443²

b² = 0.2892

b = 0.5378

=======

A = 2bh

A = 2*0.5378*0.3443

A = 0.37

======

From Fig 1, and the dimensions given, it can be seen that the distance from the circle centre to the chord is ½.

In Fig 2, the dist h' = ½, and the height of the rectangle in the smaller region is h and the half-width of that rectangle is b.

From Fig 2, we can see that b² + (h+h')² = r², where r is the radius of the circle and r=1.

The maths bit

b² + (h+h')² = r²

b² + (h+½)² = 1

b² = 1 - (h+½)²

==========

Area of rectangle is A,

A = 2bh

A = 2h{1 - (h+½)²}^(½)

dA/dh = 2{1 - (h+½)²}^(½) - 2h.½.2(h+½){1 - (h+½)²}^(-½) = 0

{1 - (h+½)²}^(½) = h.(h+½){1 - (h+½)²}^(-½)

{1 - (h+½)²} = h.(h+½)

1 - h² - h - ¼ = h² + h/2

2h² + 1.5h - 0.75 = 0

h = -1.5 ± rt(1.5² + 4.2.0.75) / 4

h = {-1.5 ± rt(8.25) }/4

h = -0.375 + 0.718

h = 0.3443

=======

b² = 1 - (h+½)²

b² = 1 - 0.8443²

b² = 0.2892

b = 0.5378

=======

A = 2bh

A = 2*0.5378*0.3443

A = 0.37

======

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