+1242 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.
+128407 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
