Bosco

Problem #1  

Get the measure of the angles on either side of PAQ   

then subtract them from 90^o to leave the size of PAQ.  

tangent of BAP is 5 / 7  =  0.7143   

tangent of DAQ is 5 / 7  =  0.7143  

note that tan^–1(0.7143)  =  35.54^o  

BAP + DAQ  =  35.54^o + 35.54^o  =  71.08^o  

PAQ  =  90^o – 71.08^o  =  18.92^o  

sin(18.92^o)  =  0.324  

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I don't have any answers, but day before yesterday I had 701 hearts, and today I have 674 hearts. 

They're a lot harder to earn nowadays; it's discouraging that they just somehow vanish into the night. 

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how many ways to make a dollar using 4 quarters 6 dimes and 8 nickels?   
The answer I got was 13 and I was just looking for confirmation.  

4 Quarters                                                       1 way  

                     4 quarters  

3 Quarters                                                        3 ways  

                     3 quarters, 2 dimes, 1 nickel 

                     3 quarters, 1 dime, 3 nickels 

                     3 quarters, 0 dime, 5 nickels  

2 Quarters                                                         5 ways  

                     2 quarters, 5 dimes, 0 nickel  

                     2 quarters, 4 dimes, 2 nickels  

                     2 quarters, 3 dimes, 4 nickels  

                     2 quarters, 2 dimes, 6 nickels  

                     2 quarters, 1 dime, 8 nickels  

1 Quarter                                                            3 ways  

                     1 quarter, 6 dimes, 3 nickels  

                     1 quarter, 5 dimes, 5 nickels  

                     1 quarter, 4 dimes, 7 nickels  

0 Quarter                                                             1 way  

                     0 quarter, 6 dimes, 8 nickels  

                                                           total          13 ways  

You found 13 ways.  I can't find a 14th one, 

so it looks like we both got the same answer. 

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How many ways can you write two different letters chosen from 
A B C D E F G H I J K L M N O P 
so that they are in alphabetical order, and the first letter is a vowel?  

two different letters  

in alphabetical order  

the first letter is a vowel  

If the first letter is A, then there are 15 possibilities.  (Second letter can be anything past A.)  

If the first letter is E, then there are 11 possibilities.  (Second letter can be anything past E.)  

If the first letter is I, then there are 7 possibilities.     (Second letter can be anything past I.)  

If the first letter is O, then there is 1 possibility.         (Second letter can be only P.)  

Add them up.                                34 possibilities.   

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Fill in the blanks to make a quadratic whose roots are $1$ and $-1.$  
x^2 + ___ x + ___    

If the roots are +1 and –1 then the factors are (x – 1)(x + 1)  

Multiplied out, the quadratic is x² – 1  

So, the first blank holds a 0 (zero) and  

the second blank holds a –1 (negtive 1)   

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cosine = adjacent over hypotenuse   

cos(65) = 5 / AC  

0.4226 = 5 / AC   

AC = 5 / 0.4226   

AC = 11.832 rounded to two decimal places is 11.83  

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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.  

A = base x height.  You know the base is 12. 

Here's how to find the height.  

Draw the rhombus, with it leaning to the right. 

Drop a perpendicular from the top left corner to the base.  

You have created a right triangle with hypotenuse 12 and angle 30^o.  

sine = opposite / hypotenuse  

sine(30) = height / 12  

0.5 = height / 12  

height = 6   

Area = base x height  

Area = 12 • 6   

Area = 72   

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I don't know if this is the classroom approved method. 

A geometric series is formed by multiplying a term by a number to find the next term. 

So, if we call that multiplyer number x, then a series would look like this. 

1st number times x  =  2nd number  

2nd number times x  =  3rd number  

3rd number times x  = 4th number  

4th number times x  =  5th number  

By this we can see that 1st number times x⁴  =  5th number  

Now that we have a principle established, let's substitute    

the specific numbers given in the above problem   

48 times x⁴  =  243/16    

                                                      48x⁴  =  243/16   

                                                          x⁴  =  243/(16)(48)  

                                                          x  =  (243/768)^1/4   

Using my scientific calculator    

to take the fourth root                        x  =  0.75  

The series is    48,  36,  27,  20.25,  15.1875  

Note that 15.1875 is the same as 243/16  

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< I can't see melody on the scoreboard >   

Click the down arrow at the bottom of the Top Users list, then look again.  

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