Bosco

One ordered pair $(a,b)$ satisfies the two equations  $ab^4 = 48$ and $a^2 = 24$.  What is the value of $b$ in this ordered pair?    

                                                                            a²  =  24    ===>    a  =  sqrt(24)

                                                                           ab⁴  =  48   

                                                              sqrt(24) • b⁴  =  48   

                                                                                             48          

                                                                              b⁴  =  ––––––––   

                                                                                          sqrt(24)    

Multiply right hand side by

sqrt(24) / sqrt(24) which is  

simply multiplying it by one.                                                   48              sqrt(24)    

                                                                              b⁴  =  ––––––––  •   ––––––––   

                                                                                          sqrt(24)         sqrt(24)   

                                                                                           48 •  sqrt(24)    

                                                                               b⁴  =   ––––––––––––     

                                                                                                   24     

                                                                                b⁴  =  2 • sqrt(24)    

                                                                                b⁴  =  sqrt(96)    

                                                                                b⁴  =  96^1/2     

                                                                                  b  =  (96^1/2)^1/4    

                                                                                  b  =  96^1/8      

From root function on scientific calculator                  b  =  1.769228    

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In a rhombus, all the side lengths are $12,$ and one of the angles is equal to $150^\circ.$  Find the area of the rhombus.     

                             Area  =  base • height    

                             Area  =  12 • 6  

                             Area  =  72  

Where did that value for the height come from?  

One of the angles is 150^o so the one next to it is 30^o.  

Drop a perpendicular from the 150^o angle at the top vertex.   

Use trig.  You know the hypotenuse is 12 and the angle is 30^o.  

sin(30) = h / 12   (opposite over hypotenuse)  

0.5 = h / 12   ====>  h = 6  

^.

Consider:         sqrt(4x + 1) / sqrt(–4x – 1)    

Not Allowed:     denominator equal to zero  

                        negative number under a radical  

Denominator:  

x has to be smaller than –1/4   

that is, larger than |1/4| in the negative direction    

to keep the number under the radical greater than zero      

Numerator:  

However, any x smaller than –1/4 creates a negative number under the radical    

Therefore, there is no solution to this problem;  

that is, there is no value of x that will make the expression well-defined.   

.

                   (21)(1/7) + (15)(–1/5)   

                           3 – 3  =  0   

Nowhere did it say that the multiple had to be an integer.   

The only problem I see with this is the question, is zero positive?

_.

Jessica has one cube with a side length of $1,$ and another cube with a side length of $1.$  She then glues the two cubes together, to form one single solid.  What is the smallest possible surface area of the single solid?  

Each cube will have a surface area of 6 square units, for a total of 12 square units.  

When Jessica sticks them together, she lost two surfaces because they're on the inside now.  

So the remaining surface area is 10 square units.  

               (AB)² • (E)   

Area  =  ––––––––––   

                    3   

The volume of a pyramid is one-third the area of the base, times the height..  

Since ABCD is a square, the area of the base is one side squared.  

_.

What is the coefficient of x^2 in (x^3 + x^2 + x + 1)(x^4 + 3x^3 + 8x + 7x + 1)?  

                                       Is that 8x a typo that should have been 8x² ?  

To get the product, you have to multiply every term inside the   

first parentheses times every term in the second parentheses.  

You could write it like this  

                                                  x³ • (x⁴ + 3x³ + 15x + 1)       (1)   

plus                                            x² • (x⁴ + 3x³ + 15x + 1)       (2)   

plus                                              x • (x⁴ + 3x³ + 15x + 1)       (3)   

plus                                              1 • (x⁴ + 3x³ + 15x + 1)       (4)   

Let's count up everything that results in an x to the second power.  

From (1) there is no x²   

From (2)                                     1 x²    

From (3)                                   15 x²    

From (4) there is no x²    

Totalling up the x² we have 16x² so the coefficient of the x² term is 16.  

_.

Catherine rolls a standard $6$-sided die six times.  If the product of her rolls is $2000,$ then how many different sequences of rolls could there have been?  (The order of the rolls matters.)   

Combinations of 2, 2, 4, 5, 5, 5 and combinations of 1, 4, 4, 5, 5, 5    

I cannot think of any other combinations that would multiply to 2000.  

The number of permutations of 2, 2, 4, 5, 5, 5 is 6! = 720   

The number of permutations of 1, 4, 4, 5, 5, 5 is 6! = 720   

So the total number of sequences would be 720 + 720  =  1440 sequences   

_.

What is the smallest positive integer that is the sum of a multiple of $15$ and a multiple of $21$? (Remember that multiples can be negative.) 

                   (15)(3) + (21)(–2)  

                           45 – 42  =  3  

              You can get it another way.  

                   (21)(3) + (15)(–4)  

                           63 – 60  =  3  

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