Bosco

Find the ordered triple (p,q,r) that satisfies the following system:  

p - 2q = 3  

q - 2r = -2 + q  

p + r = 9 + p  

I first answered this problem, jeez, it must have been at least a year ago by now.  

Since then, I've copied and pasted the answer in again a dozen times or more. 

Oh well, once more unto the breach.........  

Let's number those equations, for identification only:  

(1)  p – 2q  =  3  

(2)  q – 2r  =  –2 + q  

(3)  p + r  =  9 + p  

Per equation (2), r = 1  

Per equation (3), r = 9  

You can't have both at the same time.  The system has no solution.  

_.

Oh, I see how it works now   

                                                      area₁  x  area₂  x  area₃     

                                                     (4 x 3)  x  (4 x 2)  x  (3 x 2)    

remove parentheses                      4 x 3   x   4 x 2   x   3 x 2      

rearrange and group                     (4 x 3 x 2)   x   (4 x 3 x 2)  

                                                                   24  x  24    

That's how it happens  

_.

I don't know if it's a coincidence,   

but if you mutiply those areas                        12 x 8 x 6 = 576   

then take the sqrt of the product                    sqrt(576) = 24    

that's the same as the volume.  

_.

Find the volume of a rectangular solid if the areas of three of its adjacent faces are 12, 8, 6  

If the dimensions are 4 x 3 x 2 you get those areas.   

I can't figure out another way to get the same areas.  

So the volume would be 4 x 3 x 2  =  24 cubic units   

_.

Never mind.  I read it wrong.  I think. 

Is that supposed to bb 2^(n+1) or (2^n) + 1   ??

Joan's stamp collection consists of three books. Two tenths of her stamps are in the first book, several sevenths are in the second, and there are 303 stamps in the third book. How many stamps are there in Joan's collection?  

Ditto Sportzin #1.  2/10 in the 1st book & 7/10 in the 2nd book,

that takes care of 9/10 of the total, leaving 1/10 in the 3rd book.  

The problem is weirdly stated, and requires thought to make sense. 

_.

<<<< it appeared that the positive and negative root 17 were the minimum x values >>>>   

I approximated 1/sqrt(2) as 0.707 and used Desmos to graph y = 0.707(x² – 18)  

My parabola crossed the x-axis near +4.2 ... that's close enough to sqrt(17) that    

I'd say we got the same answer there.  

I believe, however, that that is not the shortest distance to the origin.  

If you look at the graph, you see that the curve is slanting outward and 

crosses the x-axis at an angle.  The measurement to that intersection  

from the origin would be a line on top of the x-axis itself.  

The shortest distance from the origin to the curve has to be the line that's     

perpendicular to the curve.  That's going to be somewhere below the x-axis.    

Eyeballing the graph on Desmos, which is an approximation, the point seems    

to be somewhere close to (4, –2) on the positive side.  I don't know how to do   

the calculations that would determine it accurately.   

_.

If the tax rate is set at 5%, calculate the total tax amount that a company would need to pay if they import goods worth $50,000 into the United States.   

                                      Tax  =  5% times $50,000    

                                      Tax  =  0.05 • $50,000     

                                      Tax  =  $2,500   

_.

A 25 foot ladder leans against a vertical wall. The foot of the ladder is 7 feet from the base of the wall. If the top of the ladder slips 4 feet down the wall, how far will the foot of the ladder slide?      

One way to do this is to use Pythagoras' Theorem twice.  Sort of a before and after.    

This would be much better if I could just draw the pictures.  But I can't, so here goes...   

Visualize the ladder as the hypotenuse of a right triangle.  

Then the hypotenuse is 25 and the ground side is 7.   

By the theorem a² + b² = c², then    a² (the wall)  =  c² (the ladder) – b² (the ground)        

                                                         a²  =  25² – 7²    

                                                         a²  =  625 – 49  =  576       

                                                         a  =  sqrt(576)  =  24 (the height of the wall side)      

When the ladder slips 4 feet down the wall, the new "a" becomes 20   

The hypotenuse stays the same 25 because it's a ladder   

So now we can say                         b² (the ground)  =  25² (the ladder) – 20² (the wall)         

                                                       b²  =  625 – 400  =  225    

                                                       b  =  sqrt(225)  =  15 (the length of the ground side)    

When the top of the ladder slid down the side of the wall 4 feet,

the bottom of the ladder slid out until it was 15 feet from the wall.   

The ladder was already 7 feet from the wall, so the bottom of the ladder slid 8 feet.   

_.

Our badminton team has finished 75% of its season.  So far, we have won 20% of the games we played.  What percent of the remainder of our games must we win in order to finish the season with the same number of wins as losses?    

You have won 20% of 75% of the total number of games    

in the season.  That means you've won 15% of the total    

number of games in the season.  

To achieve a 50% winning season, you will need to win   

another 35% of the total number of games in the season.  

However, there are only 25% of the total number of games   

in the season left to play.  There aren't enough games left.    

It's impossible for you to finish the season with 50% wins.    

_.