+64 A picture frame holds an 8-inch by 10-inch photograph. The frame adds a border x inches wide around 3 sides of the photo. On the fourth side, the frame is wider to accomodate a decoration on the frame. The fourth side is (3x-1) inches wide.
a. Write a quadratic expression for the combined area of the frame in terms of x.
b. If the border on three matching sides is one inch, what is the combined area of the frame?
c. If the combines area if the frame is 165 inches, find x.
+129852 a. An expression for the area is : L * W = (8 + 2x) ( 10 + x + [3x - 1]) = (8 + 2x) (9 + 4x)
b. If the border is one inch, then x = 1, and we have :
[8 + 2(1)] * [ 9 + 4(1) ] = [ 10 ] * [13] = 130 in^2
c. We need to solve this :
(8 + 2x) ( 9 + 4x) = 165 simplify
8x^2 + 50x + 72 = 165 subtract 165 from both sides
8x^2 + 50x - 93 = 0 factor
[4x + 31] [ 2x -3 ] = 0
Setting the first factor to 0 produces a negattive value for x
Setting the second factor to 0, we have that
2x - 3 = 0 add 3 ro both sdes
2x = 3 divide both sides by 2
x = 3/2 inches = 1.5 inches
