Bosco

My square patio is tiled with square tiles, all the same size.  All the tiles are gray, except the tiles along the two diagonals, which are all yellow.  (The corners are yellow, the center is yellow, and all the tiles along the diagonal in between are yellow.)  If there are $16$ yellow tiles, how many gray tiles are there?  

Think of these criteria:  "The corners are yellow, the center is yellow"   

That accounts for 5 yellow tiles.  It leaves 11 yellow tiles for "all the tiles along the diagonal in between"  

The number of tiles along each diagonal in between will be the same, so it must be divisible by four.  

11 is not evenly divisible by 4, so I believe there's something wrong with the problem as stated.  

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A member of the junior varsity swim team will be elected as President.  Each member is equally likely to be elected.  The probability that a sophomore is elected is $4$ times the probability that a freshman is elected.  Everyone on the junior varsity swim team is either a freshman or sophomore.  What fraction of the swim team are sophomores?  

As each member is equally likely to be elected, the determining   

factor would be the larger number of sophomores.  

Since the election likelihood of sophomores to freshment is 4:1,   

then there are four times as many sophomores.  

The fraction of the team that are sophomores is 4 / 5.    

Random example:  4 sophomores and 1 freshman.  

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What real value of t produces the smallest value of the quadratic t^2 -9t - 36 + 4 - 30t?  

Combine like terms                                    t² – 39t – 32   

This is a parabola that opens upward.  

The smallest value will occur at the vertex.  That's the point where the curve turns around and heads back up. 

The first derivitive expresses the slope.  At the vertex, the slope equals zero.  Setting the slope equal to y:   

                                                                y  =  t² – 39t – 32   

So the first derivitive is                             y'  =  2t – 39  

Set equal to zero                                      2t – 39  =  0 

                                                                   t  =  19.5    

The problem doesn't ask what   

the smallest value is, but you  

can find it by plugging 19.5  

into the original equation                    (19.5)² – 39(19.5) – 32  

                                                            380.25 – 760.5 – 32  =  – 412.25   

Checked answer by pasting y=t^2–39t–32 into Desmos.  

After re-reading answer, I edited one word for terminology correction.  

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Write the number "five hundredths, six thousandths, forty-two millionths" as a decimal.  

                0.05                      5 hundredths    

                0.006                    6 thousandths   

                0.0000                  0 ten-thousandths  

                0.00004                4 hundred-thousandths (same as 40 millionths)   

                0.000002              2 millionths   

              ^{––––––––––}  

                0.056042               five hundredths, six thousandths, forty-two millionths  

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x² – 2x + 1 has a repeated root.  Factor to (x – 1)(x – 1)   

or another way of saying that is (x – 1)²  and the root is 1.  

edited to add:  The root is 1 if x² – 2x + 1  equals zero.  

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In a certain regular polygon, the measure of each interior angle is $2$ times the measure of each exterior angle.  Find the number of sides in this regular polygon.     

The sum of the interior angle and the exterior angle at a vertex is 180^o.  

Since the value of the interior angle is twice that of the exterior angle, the exterior angle must equal 60^o.  

The sum of all the exterior angles of a regular polygon is 360^o irrespective of the number of vertexes.  

Since one exterior angle is 60^o there must be six of them to add up to the total of 360^o. 

Since there are the same number of sides that there are vertexes, the polygon has six sides.  

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A sector of a circle is shown below.  The sector has an area of $60 \pi.$  What is the radius of the circle?  

I'm not sure what \pi means.  Does the backslant mean divided by pi, or is the backslant just another one of those useless symbols like all those dollar signs in every problem I look at.  I don't know about you, but I would use a forward slant to mean divided by.  I'm going to treat the backslant as meaningless.  

The shaded area is a fourth of the circle, indicated by the right angle in the center. 

                                                          A_circle  

                                                        –––––––  =  60 π  

                                                             4  

                                                           A  =  240 π  

                                                           A  =  π r²   

therefore                                       240 π  =  π r²     

                                                            r²  =  240  

                                                            r  =  sqrt(240)  

                                                            r  =  4 • sqrt(15)  

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                   1• 2 • 2 • 2  =  8 ways to spell ARCH.  

Start with the A. There is only one A so 1 way to get there.    

Once you leave the A, there are 2 ways to get to an R.  

Once you leave the R, there are 2 ways to get to a C.  

Once you leave the C, there are 2 ways to get to an H.   

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