Bosco

How many isosceles triangles with whole-number length sides have a perimeter of 20 units?   

        Start with the shortest possible base, then work upwards.    

        Base      Sides   

            1          9.5     Won't work.  The problem says whole-number length sides.  

            2           9  

            4           8  

            6           7    

            8           6   

          10           5        Won't work.  Triangle Inequality Theorem, which states that    

                                    the sum of any two sides is greater than the third side.     

        Looks like the answer is going to be that there are 4 of them.   

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What real value of $t$ produces the smallest value of the quadratic $t^2 -9t - 36 + 13t - 60$?  

Combine like terms                                  t² + 4t – 96   

Arbitrarily say this equals "y"            y  =  t² + 4t – 96   

The "degree" of an equation is determined by the highest power of a variable in the equation. 

Since the highest power is the squared term, we know the curve is a parabola that opens upward.  

The smallest value of "y" is the bottom where it turns around and goes back up.  

We can find this by taking the first derivitive of the equation and setting it equal to zero.  

The first derivitive defines the slope of a curve at any point. 

So the first deritive of t² + 4t – 96 is 2t + 4  

Set equal to zero is                           2t + 4  =  0   

                                                               2t  =  –4   

                                                                 t  =  –2   

I took the equation to www.desmos.com/calculator to verify the answer.  

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Laverne starts counting out loud by ${}5$'s.  She starts with $2$.  As Laverne counts, Shirley sums the numbers Laverne says.  When the sum finally exceeds $20,$ Shirley runs screaming from the room.  What number does Laverne say that sends Shirley screaming and running?  

Laverne's           Shirley's  

Counting          Summation  

     2                           2  

     7                           9   

   12                         21    

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The product of two positive integers minus their sum is 39. The integers are relatively prime, and each is less than 20. What is the sum of the two integers?  

                                11 x 5 = 55     This is their product.  

                                11 + 5 = 16     This is their sum.  

                                 55 – 16 = 39    

I can't give an explanation.  I just tried a few until I hit it.  

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In a row of five squares, each square is to be colored either red, yellow, or blue, so that no two consecutive squares have the same color. How many ways are there to color the five squares, if there must be at least three yellow squares?  

In order not to have two yellows next to each other,  

the three yellows must occupy squares 1, 3, and 5. 

That leaves squares 2 and 4 to fill, and two colors  

to do it with. There are only four ways. 

Blue Blue     Red Red     Blue Red     Red Blue  

So, the answer is four ways.  

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Catherine rolls a standard 6-sided die six times. If the product of her rolls is 200, then how many different sequences of rolls could there have been? (The order of the rolls matters.)  

The prime factorization of 200 is 2 x 2 x 2 x 5 x 5 but  

that's only five factors and Cat rolled six times so we  

need to add a 1 to the rolls. 

She had to roll these numbers:  1, 2, 2, 2, 5, 5     

How many sequences of these could have occurred.  

The permutation of six things taken six at a time.  

                            ₆P₆ = 720

Oh, she could have rolled 1, 1, 4, 2, 5, 5  etc.      

_.

In order not to have two yellows next to each other,  

the three yellows must occupy squares 1, 3, and 5. 

That leaves squares 2 and 4 to fill, and two colors  

to do it with. There are only four ways. 

Blue Blue     Red Red     Blue Red     Red Blue  

So, the answer is four ways.  

_.

     A_{large square}  =  6.25 units²  

     The base of each triangle is 1 unit. 

     There are 2¹/₂ bases going across.  

     So the side of the square is 2¹/₂ units.  

     Two and a half squared is six and a quarter.  

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