Assignment- Finding the Largest Area

A picture framer has a piece of wood that measures 1 inch wide by 50 inches long. She would like to make a picture frame with the largest possible interior area. Complete the following activity to help her determine the dimensions of the frame that she should use to achieve her goal.

1. Use the situation shown in the figure to write an equation in x and y for the outer perimeter of the frame. (Remember that the outer perimeter will equal 50 inches.)

2. Use your equation from Question 1 to help you find the value of y for each value of x given in the table. Complete the y column of the table. (Note: The first two columns of the table give possible combinations for the outer dimensions of the frame.)

3. How is the interior width of the frame related to the exterior width of the frame? How is the interior height of the frame related to the exterior height of the frame? Use these relationships to complete the two columns of the table labeled "Interior Width" and "Interior Height."

4. Complete the last column of the table labeled "Interior Area" by using the columns of dimensions for the interior width and height.

5. From the table, what appears to be the largest interior area of the frame? Which exterior dimensions of the frame provide this area?

6. Use the patterns in the table to write an algebraic expression in terms of x for the interior width of the frame.

7. Use the patterns in the table to write an algebraic expression in terms of y for the interior height of the frame.

8. Use the perimeter equation from Question 1 to rewrite the algebraic expression for the interior height of the frame in terms of x.

9. Find a function A that gives the interior area of the frame in terms of its exterior width x. (Hint: Study the patterns in the table. How could the expressions from Questions 6 and 8 be used to write this function?)