CPhill

Note that.....on the last one.......the graph of the cable car is a straight line....thus it's average rate of change is constant = 50 yd / min

From 2 - 4 min  ....the avg rate of change  for Pat is  [200 - 125 ] / 2 =  75 / 2  =  37.5 yds / min

From 4 - 5 min.....the avg rate of change for Pat is  [225 - 200] / 2  =  25 / 2  =   12.5 yds / min

Note that Pat is getting tired......but the cable car isn't.....LOL!!!!

tan  A    will be undefned at   pi/2  +  n * pi

So    let  A  =    θ/2

So      θ /2  =  pi /2  +  n* pi        multiply through by 2

So      θ  =   pi  + n*2pi

So.....it will  be defined at all real numbers except    pi  +  2pi * n

So.... "d"  is correct...

Welcome aboard, dana.....!!!!!

"Of " means "Multiply"

So

2/3 x 3/7  =   6 / 21    =  2 / 7

We have   2 + 10 + 7 =   19 equal parts

So

One side  =  (2/19) * 228  =   24

The next side is 10 : 2 =   5 times this =  120

So...the remaining side  =  228 - 120 - 24   =  84

f(2)  =  12

The inverse to f  =    (x / 3)^(1/2)  = f ^-1

The derivative of this is    (1/3)(1/2)(x/3)^(-1/2)  = (1/6)(x /3)^(-1/2)

So   

df^-1 / dx     at     f(2)   =  12   is

(1/6) (12/3)^(-1/2)  =

(1/6) ( 4)^(-1/2)  =

(1/6) / √4 =

(1 ) / ( 6 * 2)  =

1 /12

The correct answer is C

When x   is    [ 3, 5 ]   then     l x - 4 l  ≤ 1   is true

When y is    [ 4, 8 ]  then   l x - 6 l  ≤  2    is true 

2^(x+y) * 3^(x-y) * 6^(2x+2y) = 72

2^(x + y) * 3^(x - y) * ( 6^2 )^(x + y)  =  72

2^(x + y) * 3^(x - y) * (36)^(x + y) = 72

[ 2 * 36]^(x + y) * 3^(x - y) = 72

[72]^(x + y) * 3^(x - y) =  72

If we let x, y =  1/2 we have that

2^1 * 3^0 * 6^2 = 72

2 * 1 * 36 = 72

So

(x, y) =  (1/2, 1/2)

 (2^(2+4)-2(2^n))/(2(2^(n+3)).  =

[ 2^6  - 2(2^n ] / [ 2 * 2^(n + 3) ]  =

[ 2^6 - 2^(n + 1) ] / [ 2^(n + 4) ]  =

[ 64 - 2^(n + 1) ] / [2 ^(n + 4)  ]  =

[ 2 * 2^5 - 2 * 2^n ] / [ 2* 2^(n + 3) ] =

[ 2^5 - 2^n ] / [ 2^(n + 3) ]  =

[32 - 2^n ] / [ 8 * 2^n ]   

This is all we can do

8x^2 + 15x + c = 0

To have two real roots, the discriminant must be > 0

So

15^2 -  4*8*c > 0

225 > 32c

c < 225/32

c< floor [ 225/32]   

c < 7       but     also > 0 since c must be positive

So....the product of all positve c's =   7!  =   5040