CPhill

Really need to see a pic here, pigglepoox......hard for me to see what is being described

OK.....I'll see what I can do...!!!

We have two triangles....one with a height of 5 inches and the other with a height of 14 inches

And both have  the same base = 7.5 inches

So.....the total area is just the area of both triangles =

(1/2) base ( 14 + 5) =

(1/2) (7.5) ( 19)   =  71.25 square inches

The height of the parallelogram = 8

And the width = 5

So   [ ABCD ]  =    width * height =  5 * 8   =    40 ⇒  "D"

The bottom base = top base

So....the area =   (bottom base) (height) = (6) (7) =   42 square meters

f(g(x))  =  x^2 + 2

g(f(x) )   =  (x + 2)^2

So.....we want to know

x^2 + 2  =  (x + 2)^2       simplify

x^2 + 2  = x^2 + 4x + 4    subtract x^2 from both sides

2 = 4x + 4                      subtract  from both sides

-2 = 4x                       divide both sides by  4

-2 / 4 =   x =   - 1 /2

OK....glad to help

Yep.....40

Area of triangle   =  (1/2)(base)(height =  (1/2)(5) (16)  = 40 in ^2

Let the point on the graph be   [ a, (1/√2)(a^2   - 3) ]

So....if D is minimized....then so is D^2

So we have that

D^2  =   (a - 0)^2 + [ (1/√2)(a^2   - 3)   - 0)]^2

D^2 =   a^2    + (1/2)(a^2- 3)^2             take the derivative of this and set to 0

D^2 '   =     2a  +  (a^2 - 3)(2a)   = 0

D^2 ' =     2a [  1 + (a^2 - 3 ]  =  0

This implies that either

2a = 0

a = 0

Or that

1 + a^2 - 3 = 0

a^2  - 2 = 0

a = ±√2

When   a  = 0, D^2 =   0^2    + (1/2)(0^2- 3)^2  =  9/2

When a =  -√2   ,  D^2 =    (-√2)^2 +  (1/2) [ (-√2)^2 - 3 ]^2  = 5/2

The same result will resullt when a = √2

So D  will be  √[5/2]   =  √2 * √5  / 2   =  √10 / 2

And   a + b   =   12