A rectangular picture measuring 3 in. by 7 in. is surrounded by a frame with uniform width x. Write a quadratic function in standard form to show the combined area of the picture and frame.
To find the combined area of the picture and frame, we need to determine the dimensions of the frame first.
Let's start by visualizing the rectangular picture and the frame around it.
The picture measures 3 inches by 7 inches. We can represent its length as 3 inches and the width as 7 inches.
Now, to find the dimensions of the frame, we need to consider that the width of the frame is uniform. This means that the width of the frame is the same on all sides of the picture.
Let's denote the width of the frame as "x" inches.
Now, let's calculate the dimensions of the entire frame, including the picture and the frame itself.
The length of the frame (including both sides of the picture) will be 3 inches + 2 * x inches (as there are two sides of the picture).
Similarly, the width of the frame will be 7 inches + 2 * x inches.
To calculate the area of the entire frame, we multiply the length by the width:
Area = (3 inches + 2 * x inches) * (7 inches + 2 * x inches)
Expanding this equation gives us:
Area = (3 + 2x) * (7 + 2x)
To find the combined area of the picture and the frame, we add the area of the picture to the area of the frame:
Combined Area = Area of Picture + Area of Frame
The area of the picture is simply the length multiplied by the width:
Area of Picture = 3 inches * 7 inches
Now, we can express the combined area of the picture and frame as a quadratic function in standard form:
Combined Area = 21 inches^2 + (3 + 2x) * (7 + 2x) inches^2
Simplifying this quadratic function gives us the final answer.
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length = 7 + 2x
width = 3 + 2x
area = (2x+7)(2x+3)
etc