Bosco

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.    

_.    

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     

_.    

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    

_.    

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    

_.    

 What is the coefficient of x in (x^3 + x^2 + x + 1)(7x^6 + 8x^5 - 4x^4 - 10x^3 + 12x^2 - 2x - 8)?    

Every term in the first expression is multiplied by every term in the second expression.    

Determine the terms that, when multiplied together, result in the "x" coefficient product.   

(x³ + x² + x + 1)(7x⁶ + 8x⁵ – 4x⁴ – 10x³ + 12x² – 2x – 8)    gives you   – 2x    

(x³ + x² + x + 1)(7x⁶ + 8x⁵ – 4x⁴ – 10x³ + 12x² – 2x – 8)    gives you   – 8x    

                              add them together for the final value                      – 10x    

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 is irrelevant.    

_.    

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.     

_.    

A baking club wants to form an executive committee. There are $20$ people in the baking club, including Mark. In how many ways can the baking club form an executive committee with $7$ people, including Mark?     

₂₀C₇ = 77,520    

This is called the combination of 20 items taken 7 at a time.    

Sometimes called the shortened form of "20 choose 7" but    

I prefer the first way.   

_.    

Let a1, a2, a3, ..., a10 be an arithmetic series. If a1 + a3 = -2 and a2 + a4 + a6 = 1, then find a1.    

call "d" the amount of addition between successive terms.    

starting with a₁   then  a₂  =  a₁ + d                                              =   a₁ + d     

                                    a₃  =  a₂ + d  =  (a₁ + d) + d                     =   a₁ + 2d    

                                    a₄  =  a₃ + d  =  (a₁ + d + d) + d               =   a₁ + 3d    

                                    a₅  =  a₄ + d  =  (a₁ + d + d + d) + d         =   a₁ + 4d    

                                    a₆  =  a₅ + d  =  (a₁ + d + d + d + d) + d   =   a₁ + 5d    

given       a₁ + a₃  =  – 2    

                a₁ + a₁ + 2d  =  – 2    

                2a₁ + 2d  =  – 2    

                     a₁ + d  =  – 1                                   (eq 1)    

given        a₂ + a₄ + a₆  =  1    

                (a₁ + d) + (a₁ + 3d) + (a₁ + 5d)  =  1    

                3a₁ + 9d  =  1                                       (eq 2)    

multiply (eq 1) by – 9         – 9a₁ – 9d  =  9    

add (eq 2)                             3a₁ + 9d  =  1    

                                           – 6a₁          =  10    

                                                a₁          =  – 10 / 6    

                                                        a₁  =  – 5 / 3    

_.   

Every item in the first term is multiplied with every item in the second term.    

Pick out the items that, when multiplied, result in an "x" coefficient product.    

(x3 + x2 + x + 1) • (x4 – 8x3 + 17x2 – 23x + 14)     gives you + 14x    

(x3 + x2 + x + 1) • (x4 – 8x3 + 17x2 – 23x + 14)     gives you – 23x    

add them together for the final value                                     – 9x    

_.    

A line and a circle intersect at A and B as shown below. Find the distance between A and B.    
The line is x = 4, and the equation of the circle is x^2 + y^2 = 25.    

Substitute 4 for x in the equation of the circle to find the point(s) common to both curves.    

x² + y²  =  25    

       y²  =  25 – x²    

       y²  =  25 – 4²    

       y²  =  9    

       y  =  + 3       —>>  The curves intersect at (+4, +3) and (+4, –3)    

Since x=4 is a straight, vertical line, there is no horizontal component of the distance between A and B.    

Therefore, the distance between + 3 and – 3 is 6 units.    

_.    

Every item in the first term is multiplied with every item in the second term. 

Pick out the items that, when multiplied, result in an "x" coefficient product.  

(x³ + x² + x + 1) • (x⁴ – 8x³ + 17x² – 23x + 14)     gives you + 14x    

(x³ + x² + x + 1) • (x⁴ – 8x³ + 17x² – 23x + 14)     gives you – 23x    

add them together for the final value                                     – 9x    

_.