A picture frame consists of two rectangular strips of wood, each having a width of 1 inch on all sides. If the area of the inner light gray strip is 100 \(\text{in}^2\) , then compute the area of the outer dark gray strip in square inches.

Lightning Oct 14, 2018

#2**+1 **

Let me do this one algebraically

Let the "white" area in the middle have sides of x and y

So..it's area = xy

Note that the light gray strip plus the white area will have dimensions of (x +2) and (y + 2)

[ We are adding 1 inch to each side of the white area ]

So....this area minus the white area = area of light grey strip

Algebraically, we have

(x + 2) (y + 2) - xy = 100

xy + 2(x + y) + 4 - xy = 100

2(x+ y) + 4 = 100

2(x + y) = 96

x + y = 48

Now the total arra of the figure can be represented as ( x + 4) (y + 4)

[ We are adding 2 inches to every side of the white area ]

So

Total area - area of light grey strip - white area = area of dark grey strip

(x + 4) (y + 4) - 100 - xy =

xy + 4(x + y) + 16 - 100 - xy =

4 (x + y) + 16 - 100 =

4(48) + 16 - 100 =

192 + 16 - 100 =

208 - 100 =

108 in^2 = area of dark grey strip

CPhill Oct 15, 2018