CPhill

This seems tougher than it really  is

Note....that every  non-square natural number will have an even number of divisors

For instance.....the divisors  of 8  are  1  2  4  8

However....the perfect squares always  have an odd number of divisors

For instance....the divsors of 16  are  1  2  4  8  and 16

And  the perfect squares  which have prime square roots  will  only  have 3 divisors

For instance...9   has a square root of 3 which is prime....so...it will only have 3 divisors  ⇒ 1, 3 and 9

So...the squares  < 1000  which have prime square roots are

4            289

9            361

25          529

49          841

121        961

169

So...11  natural numbers < 1000   have  only 3 divisors

Note that the sum of  the first  n  positive integers  is just

n (n + 1)  / 2

So....the sum of the first   even positive integers  would  just be twice this  =  

(n) (n + 1)  / 2  *  2  =  ( n )  ( n + 1)

For instance....the sum of the first two positive even integers  is just

2 ( 2 + 1)  =  2  * 3  = 6

So....the sum of the first 893 positive even integers is just

893  ( 893 + 1)  =

893  * 894  =

798,342

a * b   = a^2/b

a  =  10

So

a^2  = 100

So...we are looking for all the divisors of 100.....these are

1  2   4   5    10    20     25    50     100

So...x  will take on 9 values

 x = 15

The factors of 15  are

1  3  5  15

And these sum to  24

Reversing the results, we have  :

Before pressing the asterisk  key for the last time, the number displayed must have been 324

Taking the square root of this would  result in 18

Before  pressing the asterisk key for the first time, the display would have shown "81"

Taking the square root of this would have resulted in an original positive integer of "9"

So...the steps - in order -  would have resulted in :

9

81

18

324

243

I = (V/R)(1-e^t)     multiply both sides by  R / V

l (R/V)  = 1  - e^t      add e^t   to both sides and subtract  l ( R/V)  from both sides

e^t  =  1  - l (R / V)     take the natural log of both sides

Ln  e^t  = Ln ( 1  - l (R / V) )      and by a log property, we can write

t * Ln e   = Ln  ( 1  - l (R/V) )      and  Ln e  = 1...so we have

t  = Ln ( 1  - l (R/V) )

The number of people  in the first formation  is   w ( w - 7)

The number of people in the second formation is  ( w - 3) ( w  - 5)

Since the number of people in each formation is the same, we can set these equal to solve for w.....

w  ( w  -7)  =  ( w - 3) ( w - 5)

w^2  - 7w  =  w^2  - 8w + 15     subtract w^2 from both sides

-7w  = -8w  + 15       add 8w to both sides

x  = 15

So....using either of the first two expressions....the number of people in the marching band  is

 w  ( w  - 7)  =

15  ( 15 - 7)  =

15 * 8  =

120  people

P.S.  - check to see that  ( w - 3) ( w - 5)  gives the same result....

\($\left(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\cdots\right)\left(1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots\right)=1+\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}+\cdots$\)

The sum of the first series on the left hand is given by

 1 / (1 - 1/2)  =  1  / (1/2)  = 2

And the sum of the second series on the left hand side is given by :

1  / ( 1 - ( - 1/2) )  =  1 / (3/2)   = 2/3

So.....the  product of these series sums  =    2 * 2/3   = 4/3

And the common  difference for the series  on the right side is  1/x

So....the sum of the  series on the right side  is  given by :

     1 / [ 1  - 1/x]  =  1  / [ (x - 1) / x  ]  =    x  / x -1

So....setting these equal and solving for x, we have

4/3  =   x /  ( x  -1)        cross-multiply

4 ( x - 1)  =  3x

4x  - 4   = 3x

x  =  4

The  circumference of  the base  of each cone will be the arc length of one of the sectors  =

2 pi (6)  / 3  =  4 pi  inches

So....the radius, r, of each cone (in inches) is given  by :

4pi  =  2 pi * r

4  = 2r

2  = r   

And the slant height of each cone  is the radius of the original circle

And the radius of the cone and its height will form two legs of a right triangle with a hypotenuse of 6 inches  (the slant height)

So...using the Pythagorean Theorem, the cone height, h, is given by :

h  = √[ slant height^2  - radius^2 ]  =  √ [ 6^2  - 2^2 ] = √32  =  4√2  inches

This region will form a triangle

To find the height of this triangle, let us fihd the x value of the intersection of the lines....set the y's  equal

4x  - 6  = -2x + 12     add 2x to both sides, add 6 to both sides

6x  = 18     divide both  sides by  6

x  = 3         this x value will be the height of the triangle

And the y intercept of 4x  - 6  =   -6

And the y intercept  of  -2x + 12  = 12

So....the length of the base of this triangle  =  12 - (-6)   = 18

So....the area of the tiriangle is  (1/2) (base length) (height)  =  (1/2)(18)(3)   = 27  units^2

Here's a pic :

https://www.desmos.com/calculator/9svh83h2ij