a rectangular picture 8 inches by 12 inches has a frame of uniform width x inches. Write a function expressing the total area A of the picture and the frame as a function of x.
A = (2x+12)(2x+8)
multiply that out using FOIL or distributive property
(a+b)(a+c) form
To express the total area A of the picture and the frame as a function of x, we need to calculate the area of both the picture and the frame separately, and then add them together.
The area of the picture is equal to the product of its length and width. In this case, the length is 12 inches and the width is 8 inches. So, the area of the picture is:
Area of picture = Length × Width = 12 inches × 8 inches
The area of the frame is equal to the difference between the total area of the frame and the area of the picture. To calculate the total area of the frame, we have to add two times the width of the frame to both the length and the width of the picture. So, the length of the frame including the picture would be (12 + 2x) inches, and the width would be (8 + 2x) inches.
Therefore, the area of the frame is:
Area of frame = (Area of frame and picture) - (Area of picture)
= (Length of frame and picture) × (Width of frame and picture) - (Area of picture)
= (12 + 2x) inches × (8 + 2x) inches - (12 inches × 8 inches)
Finally, the total area A of the picture and the frame is the sum of the area of the picture and the area of the frame:
A(x) = Area of picture + Area of frame
= 12 inches × 8 inches + (12 + 2x) inches × (8 + 2x) inches - (12 inches × 8 inches)
Simplifying the equation further can help in obtaining a more concise expression for the total area A(x).
To calculate the total area A of the picture and the frame, you need to consider the area of both the picture and the frame separately and then add them together.
The picture has dimensions of 8 inches by 12 inches. Since the frame has a uniform width of x inches, you can calculate the dimensions of the inner part of the frame by subtracting twice the width (2x) from the overall dimensions.
The length of the inner part of the frame will be 8 inches minus 2x, and the width will be 12 inches minus 2x. Thus, the area of the picture is:
Picture area = Length × Width
= (8 - 2x) × (12 - 2x)
= 96 - 16x - 24x + 4x^2
= 4x^2 - 40x + 96
Next, the width of the frame will be x inches on two opposite sides, and the length of the frame will be x inches on the other two opposite sides. So the dimensions of the frame are 8 inches + 2x and 12 inches + 2x.
The area of the frame is given by:
Frame area = Length × Width
= (8 + 2x) × (12 + 2x)
= 96 + 16x + 24x + 4x^2
= 4x^2 + 40x + 96
Finally, to calculate the total area A of the picture and the frame, you need to add the areas of the picture and the frame together:
Total area A = Picture area + Frame area
= (4x^2 - 40x + 96) + (4x^2 + 40x + 96)
= 8x^2 + 192
Thus, the total area A of the picture and the frame can be expressed as the function:
A(x) = 8x^2 + 192