Aframe for a picture is 2 inches wide. The picture inside...

Aframe for a picture is 2 inches wide. The picture inside the frame is 4 inches longer than it is wide. See the figure. If the area of the picture is 320 square inches, find the outside dimensions of the picture frame.
(PICTURE NOT COPY)

Key Concepts

Relationships Between Dimensions

Often in geometric problems, one dimension is expressed in terms of another, such as one length being a fixed amount longer than another. Recognizing and applying these relationships simplifies the process of setting up equations and solving for the unknown variables in the problem.

Understanding Geometric Frames and Borders

This concept pertains to adding a uniform border or frame around an object, which alters its overall dimensions. The process involves recognizing that the frame adds a consistent extension to every side of the original object, and using this information to relate the internal and external measurements accurately.

Quadratic Equations

When area or other product relationships in geometric problems lead to second-degree equations, quadratic equations come into play. Solving these equations, using methods such as the quadratic formula or factoring, is essential to finding the dimensions of objects based on their area or other attributes.

Algebraic Modeling

This concept involves setting up equations that represent the relationships described in a word problem. In geometry, algebraic modeling is used to translate dimensions, lengths, and areas into mathematical expressions by defining variables and framing equations according to the given conditions.

Transcript

00:01 So in this problem we're given that a frame for a picture is 2 inches wide, and that the picture inside the frame is 4 inches longer than it is wide.

00:09 And we're given a figure, and then we're asked if the area of the picture is 320 square inches, we need to find the outside dimensions of the picture frame.

00:20 So let's draw this a little bit.

00:24 This is the picture frame.

00:27 That's what we want to find.

00:28 And there is a smaller rectangle, which is the picture, which is inside here.

00:37 Okay, so we know that this side be w, then the picture inside the frame is 4 inches longer than it is wide.

00:50 So that means that the length is w plus 4.

00:56 Okay.

00:57 So now we have the dimensions of the picture, and we want, in the end, we want the dimensions, of the picture frame.

01:08 So we can deduce the dimensions of the picture frame from what we have of the picture and of what we have of the frame.

01:17 So the frame is two inches wide, so it's two inches on each side.

01:29 So it's just if we have this picture going in here, then it's just two inches added on to each side.

01:38 So that means here we'll have w plus 4 and then plus this side and this side.

01:50 So plus 2 plus 2, which is equal to w plus 8.

01:57 And it would be the same up here.

02:00 And then similarly, we have on these sides, w plus and then 2 and 2...

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