CPhill

OK....let me see if I can spot my error....

3. Expand (a+b)^3

We only need  the 3rd row of Pascal's Triangle for the coefficients  [ note that the first row  = Row 0 ]

So....we have

1a^3  + 3a^2b  + 3ab^2  + 1b^3  =

a^3  + 3a^2b + 3ab^2 + b^3

5.Find the number of paths from  to  that pass through point  (Each step can only go up or to the right.)

Note that a set of moves that get us from  A to C  is  just  (R, R, U, U, U)

And we can choose any 2 of 5 places for R in the set and still get from A to C

So.....the total number of paths from A to C  is  C(5,2)   =  10

Likewise

A set of moves that get us from  C to B  is just (U, R,R,R )

And we can choose any 1 of the 4 places for U in the set and still get from C to B

So....the the total  paths from C to B is just  C(4,1)  = 4

And the total possible paths   =  C(5, 2) * C(4, 1)    =   10 * 4   =    40

We have the following perimeter

2x + 2y  +  pi x / 2  =   12         divide through by 2

x +  y  +   (1/4)pi x  =  6

(1 + (1/4)pi ) x  + y  =  6

y  =  6  - (1 + .25pi)x

So....the total Area , A, to be maximized is

A  =  x * y  + pi  [(1/2)x]^2/2          substituting for y, we have

A =  x * [ 6 - (1 + .25pi)x] + .25pix^2 /2      simplify

A   = 6x - (1 + .25pi)x^2 + .125pix^2

A = 6x - x^2 - .25pix^2 + .125pix^2

A = 6x - x^2 - .125pix^2

A= 6x -  ( 1 + .125pi)x^2

This might be most easily solved with a graph: https://www.desmos.com/calculator/brhi8ngjzf

The value of x that maximizes the area ≈  2.154 ft

And y = [6 - (1 + .25pi)(2.154) ] ≈ 2.154 ft

So  the rectangular part  =  2.154 ft x 2.154 ft

Corrected Answer

2.    Factor both   numerator / denominator

( x^2 - 9)   factors as ( x + 3) ( x - 3)

(x^2 - 4)   factors as ( x + 2) ( x - 2)

So we have

( x + 3) ( x - 3) (3x + 2)

__________________

(x + 2) ( x - 2) ( x - 3)

Notice that   (x - 3)  "cancels" in both the numerator and denominator

Setting this to 0, we have   that  x - 3  = 0  ⇒   x  =  3

So....we have a " hole" at x = 3

And vertical asymptotes at x = -2  and x = 2

Here's a graph :   https://www.desmos.com/calculator/g9nniltjdu

The height of the skyscraper  is given by

60( tan 53  + tan 37)  ≈  124.8 ft

Last one

Independent  for the first two

Dependent for the last

First one

Let's just evaluate     6 + 5e ^(-.1(6) )  ≈  8.744

So

18 / { 8.744)  ≈  2.1

(x + 12)^2 + (x + 7)^2 + (x + 3)^2 + (x - 4)^2 + (x - 8)^2

This simplifies to

5x^2 + 20x + 282      the minimum will  at occur at the x value   (-20) / (2 * 5)  =  -20 / 10  = -2

And the minimum value  is

5(-2)^2 + 20(-2) + 282  =

20 - 40  + 282  =

262