Bosco

We have two geometric sequences of positive real numbers:     

6,a,b and a,b,54     

Solve for a      

6               =  1st  term     

6 • r           =  2^nd  term ... this is a     

6 • r • r      =  3^rd  term ... this is b     

6 • r • r • r  =  4^th  term ... this is 54     

so                 6r³  =  54    

                       r³  =  9     

                       r    =  cube root of 9     

a  =  6 • r  =  6 • cbrt(9)     

This is about 12.48 but that's just a close approximation.   

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What is the product of the two integer values for x for which |x^2 - 25| is a prime number?     

         6² – 25  =  36 – 25  =  11     

     (–6)² – 25  =  36 – 25  =  11     

These are the only two I found.  There might be more.  Their product is  –6 • 6 = –36     

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A sphere of radius 4 inches is inscribed in a cone with a base of radius 4 inches.     

With the sphere 8" diameter, and the cone base also 8" diameter,     

I don't understand how it would be possible for the lateral surface     

of the cone to taper to an apex.     

_.     

A standard six-sided die is rolled twice. What is the probability that the product of the rolls is 1, 2, or 3?     

The number ways can you get a 1, 2, or 3 out of the number of ways the two rolls could have resulted.     

number of ways to get  1 ... 1, 1                   only 1 way     

number of ways to get  2 ... 1, 2   2, 1                  2 ways     

number of ways to get  3 ... 1, 3   3, 1                  2 ways     

total number of ways to get a desired product      5 ways     

total number of ways rolls could have landed     36 ways     

probability of rolling the desired product              5 / 36 or 13.9% if you prefer stating it that way     

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A fair coin is tossed 12 times.  How many different sequences of tosses could there have been, if the number of tosses was at least     

The number is 2¹²  =  4096     

If you toss it 1 time,   the possible sequences is  2¹ = 2 ... either H or T     

If you toss it 2 times, the possible sequences is  2² = 4 ... HH, TT, HT, TH     

If you toss it 3 times, the possible sequences is  2³ = 8 ... HHH, HHT, HTH, THH, TTT, TTH, THT, HTT     

and so on     

_.     

what is a number with a one folowd with a googleplex zerows?     

A number with a one followed by a googolplex zeros is called a googolduplex or googolplexian.     

It is written as 10^(10^(10^100)) or 10^(googolplex).     

In 2007, the U.S. population was 302 million people. In 2012, it was 314 million. What was the rate of population change per year?     

rate = change per time ....  "per" means divided by.     

rate = (314 – 302) million / (2012 – 2007) year     

rate = 12 / 5 million / year     

Want it as a decimal?        rate = 2.4 mil/yr     

_.     

                              a_n  =  a₁ + (n – 1) d     

therefore                a₁  =  a_n – (n – 1) d     

                              a₁  =  1001 – [ (50 – 1) • 6 ]     

                              a₁  =  1001 – [ 49 • 6 ]  =  1001 –  294     

                              a₁  =  707     

                              a₂  =  707 + 6     

                              a₂  =  713     

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