Bosco

Two runners start at vertices A and B simultaneously and run clockwise around the perimeter of square ABCD at the same speed. A drone flies so that it is always at the midpoint of the two runners.  What is the minimum distance between the two runners?  What is the maximum distance between the two runners?    

I don't know how to solve this, so I'm going to try to figure it out,    

starting out by assuming the length of a side of the square is 1.       

While each of the runners is at their corner, the distance is 1.000.    

Picture it ... when each runner has run a fourth of the side of the    

square, the straight line distance between them is sqrt(0.75² + 0.25²)     

I'll spare you the arithmetic, that answer is 0.791.    

Then next ... when each runner has run a third of the side of the    

square, the straight line distance between them is sqrt(0.67² + 0.33²)     

I'll spare you the arithmetic, that answer is 0.747.    

And then ... when each runner has run a half of the side of the    

square, the straight line distance between them is sqrt(0.5² + 0.5²)     

I'll spare you the arithmetic, that answer is 0.707.    

You can see that the distance decreases as the runners approach the    

midpoint of their respective sides, and then as they pass the midpoints,     

the distance will start increasing again until they reach the next corner.    

Conclusion.  The maximum distance between the runners is 1.000.    

                     The minimum distance between the runners is 0.707.    

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A line and a circle intersect at A and B as shown below. Find the distance between and A and B.    
The circle is x^2 + y^2 = 1, and the line is y = x.    

x² + y² = 1 is a circle, centered on the origin, and radius 1.    

y = x is a straight line, that passes through the origin.    

Since the straight line passes through the center of the circle,    

it creates a diameter through the intersection points A and B.    

The radius of the circle is 1, so its diameter is twice that.    

So, the distance between A & B, i.e., the diameter, is 2.    

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By the Pythagorean Theorem, AB = sqrt(16,000,000 – 9,000,000)    

AB  =  sqrt(7,000,000)    

sin C  =  sqrt(7,000,000) / 4,000  ≈  2,645.75 / 4,000  ≈  0.66    

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What is the sine of an acute angle whose cosine is \frac{5}{7}?    

cosine is adjacent over hypotenuse    

given that the adjacent side is 5 and the hypotenuse is 7    

by Pythagoras, the opposide side is sqrt(7² – 5²) = sqrt(24)    

sine is opposite over hypotenuse    

so the sine is sqrt(24) / 7  =  2 • sqrt(6) / 7  or  ≈ 0.6999    

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Find the number of bases n \ge 2 such that 100_n + 1_n is prime.    

This is another way of saying    

How many numbers have the property that (n² + 1) is prime?    

They've been asking that question for ages.    

No one has ever been able to answer it, yet.    

I think mathematicians say that it is infinite.     

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We cut a regular octagon ABCDEFGH out of a piece of cardboard.    

If $AB = 1$, then what is the area of the octagon?    

There's a formula for the area of a regular octagon in terms of its side "s" 

A  =  2s² • (1 + sqrt(2))    

A  =  (2)(1²)(1 + 1.414)    

A  =  (2)(2.414)    

A  =  4.828     same as Chris ... I knew it would be.    

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All eight vertices of a cube are on a sphere (i.e. the cube is inscribed in the sphere).   A sphere is also inscribed in the cube.  If the side length of the cube is 1, then what is the radius of the sphere inside the cube?    

Ignore the outside sphere.    

Consider the sphere inscribed inside the cube.    

The sides of the cube are tangent to the edge of the sphere.    

That makes the diameter of the sphere 1 unit.    ==>>    radius = 0.5     

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The surface area of a sphere is numerically equal to twice its volume. What is the radius of the sphere?    

A = 4 π r²     V = (⁴/₃) π r³    

4 π r²  =  (2)(⁴/₃) π r³    

divide both sides by 4 π r²    

       1  = (²/₃) r    

        r  =  ³/₂     

        r  =  1.5     

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In rhombus ABCD, AC = 14 and AB = 20. Find the perimeter of ABCD.    

The perimeter of a rhombus in terms of diagonals "p" and "q" is P = 2 • sqrt(p² + q²)    

P = 2 • sqrt(14² + 20²)    

P = 2 • sqrt(196 + 400)    

P = 2 • sqrt(596) = 48.826 = 48.8    

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The area of a trapezoid is $80$. The length of one base is $16$ units greater than the other base, and one base of the trapezoid is $12$. Find the length of the median of the trapezoid.    

M = [ (12) + (12 + 16) ] / 2    

M = 40/2    

M = 20    

The median of a trapezoid is the average of its two bases.    

In this particular problem, the trapezoid's area is irrelevant.    

_.    

A line and a circle intersect at A and B as shown below. Find the distance between and A and B.    
The circle is x^2 + y^2 = 1, and the line is y = x.    

x² + y² = 1 draws a circle with radius 1, centered at (0,0).    

y = x draws a straight line through (0,0).    

Since the line goes through the center of the circle, it creates a diameter.    

Since the circle has radius 1, its diameter, the distance between A & B = 2.    

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The area of a trapezoid is $80$. The length of one base is $16$ units greater than the other base, and one base of the trapezoid is $12$. Find the length of the median of the trapezoid.    

M = [ (12) + (12 + 16) ] / 2    

M = 40/2    

M = 20     

The median of a trapezoid is the average of its two bases.    

In this problem, the area of the trapezoid was irrelevant.    

_.    

Find a six-digit multiple of $64$ that consists only of the digits $2$ and $4$.    

64 • 3,816 = 244,224     

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There are four cities: Capital City, Little Village, Mytown, and Yourtown.  The distance from Capital City to Little Village is $5$ miles.  The distance from Capital City to Mytown is $2$ miles.  The distance from Mytown to Yourtown is $3$ miles.  The distance from Little Village to Yourtown is $4$ miles.  Find the distance from Capital City to Yourtown, in miles.    

Little Village ----- 4 ----- Your Town ----- 1 ----- Capital City ----- 2 ----- My Town    

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One root of the quadratic equation $x^2 + bx -28 = 4x + 14$ is $-7$. What is the value of $b$?    

x² + bx – 28 = 4x + 14    

Change that "bx" to "kx" because I have another use for "b".    

Arrange terms in standard quadratic format ax² + bx + c = 0     

x² + kx – 4x – 42 = 0     

x² + (kx – 4x) – 42 = 0   

x² + (k – 4) x – 42 = 0      and it's given that one root is – 7    

Since one root is – 7, the other root has to be + 6 to get that – 42     

so, factored, we have (x + 7)(x – 6) which multiplied out is x² + x – 42            

& so, (k – 4) x = x which means that k = + 5 or, reverting to the words of the problem, b = 5    

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