Question

9) A 5 inch by 7 inch picture has a frame surrounding

it on all four sides of width w. The total area of the
picture frame and picture is 80 square inches.
a) Using the diagram and information, write an
expression for the total length, including the
frame, and with an expression for the total
width, including the frame.
Length expression:
Width expression:
b) Using the expressions for total length and width, write an equation for the total
area of the rectangular picture and frame (Hint: Area = l × w).
c) Using your knowledge of factoring, solve the equation from (b) for the width of
the frame surrounding the picture.

Answer 1

w = 3/2

Explanation:

Area of a Rectangle

The 5-inch by 7-inch picture has a frame of width w surrounding all four sides.

a)

The total length of the framed picture is 7 + 2w

The total width of the framed picture is 5 + 2w

b)

The total area of the framed picture is:

At = (7 + 2w)(5 + 2w)

c) We know the total area is 80 square inches, thus:

(7 + 2w)(5 + 2w) = 80

Operating:

`35 + 14w + 10w + 4w^2=80`

Simplifying:

`4w^2 + 24w -45 =0`

We now write:

`4w^2 + 30w - 6w -45 =0`

Factor out 2w from the first two terms and -3 from the last two terms:

`2w(2w + 15) - 3(2w + 15) =0`

Factor out 2w+15

(2w + 15)(2w - 3)=0

The solutions are:

w = -15/2

w = 3/2

The only feasible solution is the positive value, thus

w = 3/2