SOLUTION: you are making a scrapbook out of pictures that are 7 inches long and 5 inches wide,you want to place a border of equal width on each edge a)wrie a polynomial that represents th

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: you are making a scrapbook out of pictures that are 7 inches long and 5 inches wide,you want to place a border of equal width on each edge a)wrie a polynomial that represents th      Log On


   

Question 420805: you are making a scrapbook out of pictures that are 7 inches long and 5 inches wide,you want to place a border of equal width on each edge
a)wrie a polynomial that represents the total area of the picture wtih the border
b)find the total area of the picture with border when the wodth of the border is 2 inches
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Drawing a diagram will help you understand this.
  • Draw two rectangles, one inside the other. The distance between the inner rectangle and the outer rectangle should be the same all the way around.
  • >The inside rectangle is the picture. Label the length and width of the picture with a 7 and a 6 respectively.
  • The border is the area between the sides of the picture and the sides of the outer rectangle. As stated earlier, the width of the border should be the same all the way around. Label the width as "x".

With this diagram we should be able to figure out the dimensions of the outer rectangle (the combined picture and border). Looking at the diagram we should be able to see that the length of the outer rectangle will be an "x" plus the length of the inner triangle plus another "x" on the other end. So the length will be:
x + 7 + x
which simplifies to:
2x + 7
For the width of the outer rectangle we should see an "x" plus the width of the inner rectangle plus another "x" on the other end:
x + 6 + x
which simplifies to
2x + 6

Since the area of any rectangle is its length times its width, the area of this outer rectangle will be
A = (2x+7)(2x+6)
which, using FOIL, simplifies as follows:
A = 2x*2x + 2x*6 + 7*2x + 7*6
A = 4x%5E2+%2B+12x+%2B+14x+%2B+42
A = 4x%5E2+%2B+26x+%2B+42
This is polynomial which represents the area of the outer rectangle (the picture and border combined).

For part b we just make the "x" a 2:
A = 4%282%29%5E2+%2B+26%282%29+%2B+42
which simplifies as follows:
A = 4(4) + 26(2) + 42
A = 16 + 52 + 42
A = 68 + 42
A = 110
Since "x" is in inches, the area of the picture and border, with a border of 2 inches, is 110 square inches.