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MATH 012 − 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. (cont’d)

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Finding the Largest Area (cont’d)

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?)

Frame’s Interior Dimensions

x y Interior Width

Interior Height

Interior Area

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5

Frame’s Interior Dimensions

x y Interior Width

Interior Height Interior Area

9.0 9.5

10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0

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