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 glass piece?

Key Concepts

Quadratic Equations

The process of equating the difference between the areas of two squares to a given value leads to a quadratic equation. Solving this quadratic equation is essential, as it ultimately yields the dimensions of the square, by accounting for the squared nature of the area calculation.

Algebraic Modeling

Algebraic modeling involves converting geometric relationships into algebraic equations. By assigning a variable to the unknown length of the glass piece and expressing both the inner and outer areas in terms of this variable, one can establish an equation that represents the given conditions.

Area of a Square

The area of a square is determined by squaring the length of one of its sides. This fundamental concept allows us to compute the area of both the inner square (the glass) and the outer square (the entire frame), which is key to setting up the problem.

Composite Figures

Composite figures are shapes that are made up of two or more simple geometric figures. In this problem, the overall frame consists of an outer square from which the inner square (glass) is removed, making the problem a matter of finding the difference between these areas.

Transcript

00:01 Hello, today we are continuing step two of a part of a three -step problem.

00:07 And in this part of the problem, we want to find the dimensions of the inside part of the mirror, which is represented by this red square, which has the dimensions x by x.

00:20 In the last part of this problem, we found that the equation, to represent this area right here, was x to the fourth minus 7x square and minus 18.

00:31 So if we solve 4x, then we will get the x values that we can plug in and find the dimensions here.

00:40 So since we have a polynomial that is the fourth power, then we need some knowledge that we can write polynomials in quadratic form.

00:55 And that quadratic form a u squared plus b u plus c where u is equal to x squared so let's go ahead and write our equation first into quadratic form and when we do that we get x squared minus 7 x squared minus 18 equals zero and so we've seen that we've written this into the quadratic form here where u is x squared and so since this is now written in quadratic form we can kind of factor this equation but with instead of usually when we factor we have an x plus a number and an x plus another number this time we'll have a factor that is x squared plus a number and x squared plus another number so when we factor, we get that this is x squared.

02:09 So they need to multiply together to get this number and add together to get this.

02:15 So x squared minus 9 times x squared plus 2 equals 0.

02:29 And so when we do that, we have x squared minus 9 equals 0, and x squared plus 2 equals 0...

Please give Ace some feedback