DESIGN. For Exercises 36-38, use the following information....

DESIGN. For Exercises $36-38,$ use the following information. Jill is designing a picture frame for an art project. She plans to have a square piece of glass in the center and surround it with a decorated ceramic frame, which will also be a square. The dimensions of the glass and frame are shown in the diagram at the right. jill determines that she needs 27 square inches of material for the frame. What are the dimensions of the frame?

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Key Concepts

Area of a Square

The area of a square is determined by squaring the length of one of its sides. This concept is fundamental in geometry and is used to compute the areas of both the inner and outer squares in frame design, helping to relate the dimensions to the total area available.

Difference of Areas

In design problems that involve a frame surrounding a central object, the material used for the frame can be found by subtracting the area of the inner square from the area of the outer square. This concept is crucial when determining the additional area needed for borders or frames.

Setting Up Equations

Formulating an equation from a given physical scenario involves assigning variables to unknown quantities and relating them based on the geometric properties of the shapes involved. This process is essential in translating design problems into solvable mathematical expressions.

Solving Quadratic Equations

Many geometric design problems lead to quadratic equations when determining unknown dimensions. Being able to solve these quadratics—via factoring, completing the square, or the quadratic formula—is an important skill for arriving at the correct measurements within design contexts.

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