Bosco

The number 100 can be written as the sum of a 1-digit prime number and a 2-digit prime number. What is the product of these prime numbers?  

The 1-digit prime number      3    

The 2-digit prime number    97     

Their product                       3 • 97  =  291    

_.

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've played 75% of the season.  So, you've played   

75% of the total number of games in the whole season.   

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 that 50% winning season, you will need to win   

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

However, since there's only 25% of the season left, then   

there are only 25% of the total number of games in the  

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

You can't achieve a 50% winning season this year.    

_.

In square ABCD, P is on BC such that BP = 4 and PC = 1, and Q is on CD such that DQ = 1 and QC = 4. Find sin angle PAQ   

I wish I could post a picture on here, but I can't.   

You draw the picture, and I'll provide the words.    

After drawing lines AP and AQ, you'll see that    

you've divided the square into three sections.   

Two of the sections are right triangles.  You can     

use arctan to determine angles BAP and DAQ.     

Add them together and subtract the total from 90.    

That gives you the size of angle PAQ.      

BAP  =   tan^–1 (4/5)  =  tan^–1 (0.8000)  =  38.66^o    

DAQ  =  tan^–1 (1/5)  =  tan^–1 (0.2000)  =  11.31^o     

                                                    total      49.97^o   

Angle PAQ  =  90^o minus 49.97^o  =  40.03^o     

I did some rounding on those arctan values, so   

my final answer might not agree exactly with the    

answer in the back of your math textbook.  Allow     

me just a wee bit of wiggle room on that, please. 

_.

Catherine rolls a standard 6-sided die six times. If the product of her rolls is 2500 then how many different sequences of rolls could there have been? (The order of the rolls matters.)     

If you go to https://web2.0calc.com/questions/help-plz_51258

you can check out the earlier answer I gave, and GA's explanation     

about how to have solved the problem using permutations.    

The answer is 45 sequences, by the way, but go ahead and check       

out the posting linked above anyway, to see how it was determined.    

_.

Catherine rolls a standard 6-sided die six times. If the product of her rolls is 2500 then how many different sequences of rolls could there have been? (The order of the rolls matters.) 

If you go to https://web2.0calc.com/questions/help-plz_51258

you can check out the earlier answer I gave, and GA's elaboration   

about the way to have done it using permutations.   

The answer is 45 sequences, by the way, but check out the above   

anyway, to see how it was determined.    

_.

Could anyone also add an explanation with their answer?  

The sum of any two sides of a triangle must be greater than its third side.   

That's the explanation.  The length of the straws wasn't visible in the post.   

If x is a real number such that 2^(2x+3)=14, find 2^x    

In the problem we have 2^{(2x+ 3)} = 14   

So:  To what power do we raise 2 in order to produce 14.  

The logarithm of a number is the power to which a particular  

base must be raised to produce a specific number.  

Example:    2³  =  8         so  log₂(8) = 3   

                10²  =  100     so  log₁₀(100) = 2  

The calculator in my iPhone has a button that will calculate logs.   

So, log₂(14) = 3.8073549 and the decimal fraction keeps going  

but that much of it will get close enough for our purpose.   

So 2^(3.8073549)  =  14 

This exponent is the value needed for the exponent in the problem.  

2^(2x+3) = 14    therefore  2x + 3  =  3.8073549 

                                       2x        =  0.8073549    

                                         x        =  0.4036775   

                                         2^x  =  2^(0.4036775)   

                                         2^x  =  1.32288   

_.

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've played 75% of the season.  So, you've played   

75% of the total number of games in the whole season.   

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 that 50% winning season, you will need to win   

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

However, since there's only 25% of the season left, then   

there are only 25% of the total number of games in the  

season left to play.  You don't have enough games left.    

You can't achieve a 50% winning season.    

But there's always next year.  

_.