Activity
Here is a diagram that shows the picture with a frame that is the same width all the way around.
The picture is 7 inches by 4 inches. The frame is created from 10 square inches of framing material (in the form of a rectangle measuring 4 inches by 2.5 inches).
Write an expression to represent the length of the framed picture on its shorter side. Use to represent the width of the frame.
Enter an expression to represent the length of the framed picture on its shorter side.
Compare your answer:
Write an expression to represent the length of the framed picture on its longer side. Use to represent the width of the frame.
Enter an expression to represent the length of the framed picture on its longer side.
Compare your answer:
Use the expressions from the previous exercises to write an equation to represent the relationship between the measurements of the framed picture and its total area. Be prepared to explain what each part of your equation represents.
Enter an equation to represent the relationship between the measurements of the framed picture and its total area.
Compare your answer:
or (or equivalent)
What would a solution to this equation mean in this situation?
Compare your answer:
A solution would represent the width of the frame when all of the framing material is used.
Video: Formulating a Quadratic Equation to Represent the Model
Watch the following video to learn more about formulating a quadratic equation to represent the model.
Self Check
Additional Resources
Formulating a Quadratic Equation to Represent the Model
Let’s look at a similar situation to the previous activities but using a picture with different measurements.
Imagine a picture 8 inches by 6 inches. The frame is created from 32 square inches of framing material. The frame will be uniform in width, and all of the framing material is used.
How do we write an equation representing the total area of the framed picture?
Let’s break down the measurements that we know.
Let represent the thickness of the frame in inches.
Since the picture is 8 inches by 6 inches, it has an area of 48 square inches.
The frame is created from 32 square inches of framing material, and we must use all of it. So, the total area of the framed picture is , or 80 square inches.
The length of one side of the framed picture is .
The length of the other side of the framed picture is .
So, the area of the framed picture is . We can set this equal to 80 square inches.
Let’s multiply.
So, the equation representing the total area of the framed picture is .
This equation is a quadratic equation. A quadratic equation is defined as one that can be written in the form , where is not 0.
Try it
Try It: Formulating a Quadratic Equation to Represent the Model
A picture measures 9 inches by 7 inches. Its frame is created from 36 square inches of framing material. The frame will be uniform in width, and all of the framing material is used.
Write an equation representing the total area of the framed picture.
Here is how to find the quadratic equation to represent the total area of the framed picture:
The area of the picture is 63 square inches. The total area of the framed picture will be , or 99 square inches.
The sides of the framed picture are represented by and .
So, the total area of the framed picture is represented by .