Bosco

A rectangle in the coordinate plane is shown below.  A line with slope 2 splits this rectangle into two regions with equal area.  What is the y-intercept of this line?  The rectangle has coordinates (-1,-1), (-1,1), (1,1), and (1,-1).     

The coordinates draw a square centered on the origin.     

Any line that bisects a square goes through its center.     

The bisecting line goes through the center of the square.     

The y-intercept is (0,0) which is the center of the square.     

No matter what the slope of that line was, if it bisects the     

square, it goes through the center.of that square.     

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I have $4$ different mathematics textbooks and $3$ different psychology textbooks. In how many ways can I place the $7$ textbooks on a bookshelf, in a row, if all the psychology textbooks are together?     

Strap the psychology books together and consider it as a single unit.   

Along with the 4 math books, and the strapped books we have 5 items.     

Five items can be grouped 5P5 ways, aka 5 factorial, which is 120 ways.     

But wait.  The psychology books could have been grouped several different ways.     

In fact, the psych books can be grouped 3P3 ways, aka 3 factorial, which is 6 ways.     

In total, keeping the psych books together, the books can be placed (120) • (6) = 720 ways.     

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In a store window, there was a box of berries having a total weight of 200 kg. The berries were 98% water, by weight. After two days in the sun, the water content of the berries was only 95%, by weight. What was the total weight of the berries after two days, in kg?     

You're starting with 200 kg of berries, and 98% of that is water.     

Since they're 98% water, that means that there's 2% of solid stuff.     

2% of 200 kg means that there are 4 kg of solid stuff.     

Some of the water evaporates in the sun, but the solid stuff doesn't.     

So, after the evaporation, there still remains the same 4 kg of solid stuff.     

Since the new water content is 95%, that means that now the solid stuff is 5%.     

So 2 kg (the solid stuff) is 5% of the total weight.     

That means the new total weight is 40 kg.     

_.     

Complete the square: x^2 - 10x + 21 + 3x^2 - 2x + 8.     

                                    x² – 10x + 21 + 3x² – 2x + 8     

first, combine like terms so     

you can see what you've got        4x² – 12x + 29     

"semi-factor" it, set it up so     

you can see what you need          (2x –  n  )(2x –  n  )     

(2 • n • 2x) has to be 12x      

so n = 3                                          (2x – 3)(2x  – 3)     

by itself, that works out to               4x² – 12x + 9     

but it was 29 and that     

other 20 has to come     

from somewhere... add it                4x² – 12x + 9 + 20          

so you end up with                          (2x – 3)² + 20     

I think.  I'm not accustomed to completing the square     

without an equals sign making it part of an equation.     

_.     

     Find all solutions x to the equation 5(2x + 1) + 5*2 = 5*x + 5(x + 3).     

                                  5(2x + 1) + 5*2  =  5*x + 5(x + 3)     

     multiply it out         10x + 5   + 10   =   5x  +  5x + 15     

     combine like terms          10x + 15  =  10x + 15     

     There isn't enough time left in the universe to write down all the solutions,     

     because there are infinite solutions to an equation that amounts to x = x.     

     Every x satisfies the given equation.     

_.     

Let S be the set {1, 2, 3, \dots, 10, 11, 12}.     

How many subsets of the set S have no two consecutive primes as members?     

I don't know how many subsets exist in set S, but the only consecutive primes in the set are 2 and 3.     

So figure up all the subsets, then eliminate the ones that contain 2, 3.  BTW, 1 is not a prime number.   

_.      

     In the diagram below, we know \tan \theta = \frac{3}{4}. Find the area of the triangle.     

     The triangle is right, and the hypotenuse is 80.     

     Area of a triangle equals half the product of the base times the height.     

     A  =  (1/2)(48)(64)     

     A  =  1536     

     Recognize that 5 is a factor of 80.  And that 3, 4, 5 is a Pythagorean triplet.     

     So, just multiply all the way across by 16 and you get a 48, 64, 80 right triangle.     

     You know that the legs are 48 and 64 because the longest one is the hypotenuse.     

_.     

A G.P has its 3rd term and 7th term as 45 and 3645 respectively. what is the product of its 2nd and 4th     

Call the common ratio "r"     

1st term     

2nd term    45 / r      

3rd term     45                              45     

4th term     45 • r     

5th term     45 • r • r     

6th term     45 • r • r • r     

7th term     45 • r • r • r • r        3645     

8th term     45 • r • r • r • r • r     

Taking what we know, then         45 • r⁴  =  3645     

                                                           r⁴  =  3645 / 45  =  81     

                                                           r¹  =  3     

4th term is  45 • 3  =  135     

2nd term is 45 / 3   =   15             Ha ha, I bit.  It wasn't even necessary to solve for r.     

                                                     No matter what r is, it cancels out in the 2nd and 4th.     

product of 2nd times 4th  =  2025     

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