1a A large square carpet is initially x inches long and x in

1a.)

A large square carpet is initially x inches long and x inches wide. Randy decides to trim 3 inches off each edge of the carpet because the edges are worn. Later he goes back and trims 4more inches off each edge.

(a) Express the final area (A) of the carpet in square inches (after both trim jobs) as a polynomial function of x.
A(x) =  

(b) How many square inches of material were removed in the first trim job? Express the area removed in the first trim job (T₁) as a function of x.
T₁(x) =  

(c) How many inches of material were removed in the second trim job? Express the area removed in the second trim job (T₂) as a function of x

T₂(x) =

1d.)

Let P = (x, y) be a point on the graph of y = x² - 6.

(a) Express the distance d from P to the point (0, -1) as a function of x.
d(x) =  


(b) What is d if x = 0?
d =  

(c) What is d if x = -1?


(d) Use a graphing utility to graph d = d(x).

(e) For what values of x is d smallest?
x = (smaller value)
x = (larger value)

Solution

Area of original carpet = x*x

when 3 inch trimmed from edges we get ( x-3) and (x -3)

So, ( x-3) and (x-3) are length and width of carpet

Again 4 inches are trimmed off :

(x -3 -4) and ( x-3 -4)

So, final lenght and width of carpet (x -7) and ( x-7)

Area , A(x) = ( x-7)*( x-7) = ( x-7)^2 = x^2 -14x +49

b) In first trim job , from each corner area trimmed = x^2 - (x-3)(x-3)

= x^2 -(x^2 +9 -6x)

T1(x) = 6x -9

c) In second trim jobb , total area trimmed off = Area after 1 st trimming - Area after second triiming

= (x-3)^2 - (x -7)^2

T2(x) =x^2 -6x +9 - ( x^2-14x +49)

= 8x -40