You have been requested to display your family portrait in the lobby of the new Hampton inn in west Jordan it portrait measures x inches one side. and the other side is 2" longer than the other. the frame that surrounds the portraits is 2 inches wide.
You are planning on covering the frame with gold leaf to make it much brighter and so it can look better when it is displayed.
First: write the algebraic equation that will describe the area of the frame that needs to be covered.
Second: calculate how much gold leaf will you need if the shortest side of the portrait is 8 inches.
Start with this
portrait side 1=x
portrait side 2=x+2
frame=2 in
width of portrait --- x
length of portrait -- x+2
width of whole thing --- x+4
length of whole thing --- x+2 + 4 = x+6
area of portrait = x(x+2)
area of whole thing = (x+4)(x+6)
area of frame = (x+4)(x+6) - x(x+2)
= x^2 + 10x + 36 - x^2 - 2x
= 8x + 36
so when x=8 .........
First:
Let's define the length of the shortest side of the portrait as "x" inches. Given that the other side is 2 inches longer than the shortest side, the length of the longer side can be represented as "x + 2" inches.
The actual dimensions of the portrait will be increased by the width of the frame, which is 2 inches on each side. So, the dimensions of the framed portrait will be:
Length = (x + 2) + 2 + 2 = x + 6 inches
Width = x + 2 + 2 + 2 = x + 6 inches
The area of the frame that needs to be covered with gold leaf can be calculated by subtracting the area of the portrait from the area of the framed portrait. The area of the framed portrait is the product of its length and its width, while the area of the portrait is the product of its original length and its original width.
Area of the frame = (x + 6) * (x + 6) - x * (x + 2)
Simplifying the equation, we get:
Area of the frame = x^2 + 12x + 36 - x^2 - 2x
Area of the frame = 10x + 36
Second:
We are given that the shortest side of the portrait is 8 inches (x = 8). Substituting this value into the equation:
Area of the frame = 10(8) + 36
Area of the frame = 80 + 36
Area of the frame = 116 square inches
Therefore, you would need 116 square inches of gold leaf to cover the frame if the shortest side of the portrait is 8 inches.