Bosco

Find the number of arithmetic sequences such that:    
* The arithmetic sequence contains three terms    
* All the terms are integers in \{0, 1, 2, \dots, 10\}    

* The sum of the terms is 9.     

The ones I find are     0, 3, 6  and of course  6, 3, 0

                                   1, 3, 5  and  5, 3, 1     

                                   2, 3, 4  and  4, 3, 2     

                                   3, 3, 3  if it counts

I can't think of any more.  I found seven of them.    

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In \triangle PQR, we have \angle P = 90^\circ, PQ=3, and QR = 5. Find \tan R.   

By Pythagoras  PR² = QR² – PQ²    

                          PR  =  sqrt(25 – 9)  =  4     

                       tan R  =  opposite over adjacent  =  3 / 4      

                       tan R  =  0.7500    

If you draw / label the triangle, the answer becomes clear.    

I'd like to show you, but I don't know how to post a picture.   

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The area of a cross section of a sphere is 64\% of the largest possible cross sectional area of the sphere. If the sphere has radius 1/2, what is the area of the cross section?    

A cross section of a sphere would be a circle, right?  The largest circle would be through the center, so ...     

A_{through center}  =  π r²  =  3.1416 • 0.5²  =  3.1416 • 0.25     

                       =  0.7854    

 0.7854 • 0.64  =  0.5027 unit²    

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Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of triangle ABC?    

A is at (0,0), B is at (1,0), and C is at (1,1).   

Those three points make a right triangle.    

The base   is (0,0) to (1,0)  =  1 unit    

The height is (1,0) to (1,1)  =  1 unit    

The area of a triangle is (1/2) • base • height  =  (1/2) • 1 • 1  =  (1/2) unit²    

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A and B are two points on a sphere of radius 2. We know the space distance between A and B is 2, What is distance from A to B along the (minor) arc of a great circle?    

Consider the lines from the center of the sphere to A and to B.    

Each of these two lines is the radius of the sphere, namely, 2.    

Since the distance from A to B is 2, an equilateral triangle is formed.    

Every angle of an equilateral triangle is 60^o which is 1/6 of the great circle.    

The circumference of the great circle is π d  =  3.1416 • 4  =  12.5664.    

1/6 • 12.5664  =  2.0944 is the distance from A to B along the curvature of the sphere.    

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A basket contains 7 green marbles and 6 red marbles. Two marbles are taken from the basket at random without replacement (i.e., the first marble is not put back before the second marble is drawn). What is the probability that the first marble is green, and that the second marble is red?    

Prob 1st marble green is   7 / 13    

Prob 2nd marble red is      6 / 12    

Prob both happening is     (7 / 13) x (6 / 12) = 42 / 156  reduces to  7 / 26    

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Simplify (x^4 + x^3 + x^2 + x + 1) + (x^4 - x^3 + x^2 - x + 1).       

stack them for addition    

  x⁴  +  x³  +  x²  +  x  +  1    

  x⁴  –  x³  +  x²  –  x  +  1    

 2x⁴           + 2x²         + 2   =   2(x⁴ + x² + 1)    

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(x+2) / (x)  •  (x+3) / (x+1)  •  (x+4) / (x+2)  =  10    

cancel the (x+2) ' s   

(1) / (x) • (x+3) / (x+1) • (x+4) / (1) = 10    

[ (1) • (x+3) • (x+4) ] / [ (x) • (x+1) • (1) ] = 10    

(x² + 7x + 12) / (x² + x) = 10    

x² + 7x + 12 = 10x² + 10x    

0 = 9x² + 3x – 12              this will factor (it would reduce first, but I'm skipping that step)   

(3x + 4) • (3x – 3) = 0    

                      3x + 4 = 0     this x will not produce integer numerators and denominators, so disregard    

                      3x – 3 = 0    

                            3x = 3    

                              x = 1     therefore (x+2) / (x) = 3 / 1 = 3 is the value of the first fraction   

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what is the answer 65x+17-98=-4    

given that                                            65x + 17 – 98  =  – 4     

Take all the constants to one side,    

by convention the right-hand side       65x  =  – 4 – 17 + 98    

                                                             65x  =  77    

                                                                x  =  77 / 65    

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