Bosco

tangent = opposite over adjacent  

tan(80^o)  =   pole / 22  

5.671  =  pole / 22  

pole  =  (5.671)(22)  =  124.74 rounded to a foot  =  125  

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cosecant = hypotenuse / opposite  

hyp = 25  

opp = 20  

So the adjacent side = square root of [ (25)² – (20)² ] = sqrt(225)     

adj = 15  

cotangent = adjacent / opposite  

cot = 15 / 20  

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A chain with two links is 13 cm long. A chain made from three links of the same type is 28 cm long. How many centimeters are in the length of a chain made from 25 such links?    

Let's take a run at this improbable-looking problem.  

Start with a 2 links chain, which is 13 cm long.   

Add one link, you get a 3 links chain, 28 cm long.    

We note that adding one link adds 15 cm to the length. 

Therefore, adding 23 links should add 23 x 15 = 345 cm to the length.  

The original 2 links, which is 13 cm long  

plus another 23 links, which is 345 cm long   

gives us a 25 links chain, which is 358 cm long.  

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At Hamburger Heaven, it costs 4.25 for a burger and a drink, 4.50 for a burger and fries, and 7.80 for two burgers and a drink. How many dollars would it cost for one burger? Express your answer to the nearest cent.   

2B + D  =  7.80      per third phrase  (equation 1)   

  B + D  =  4.25      per first phrase   (equation 2)    

    B        =  3.55      subtracted (2) from (1)    

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What is the 2009th term in the arithmetic sequence whose first four terms are 2, 10, 18, 26?      

Let's see if we can get a little pattern going here 

We can see that the increment is 8, so, tabulating    

Term

Number (n)         Value    

         1                 2 + (8)(1 – 1)    

         2                 2 + (8)(2 – 1)    

         3                 2 + (8)(3 – 1)    

         4                 2 + (8)(4 – 1)    

         n                 2 + (8)(n – 1)    

   2009                 2 + (8)(2009 – 1)    

The 2009^th term therefore is 16,066    

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Let $a_1, a_2, a_3,\dots$ be an increasing arithmetic sequence of integers. If $a_4a_5 a_6 = 6$, what is $a_3a_7$?      

Do you mean a₄ + a₅ + a₆ = 6 ?    

If so, I think that would be 1 + 2 + 3 = 6    

which would make a₃ = 0 and a₇ = 4     

if I understand the problem correctly.   

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How many integers belong to the arithmetic sequence 13, 20, 27, 34, $\dots$, 97?    

13, 20, 27, 34, 41, 48, 55, 62, 69, 76, 83, 90, 97    I count 13 of them.   

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