A picture frame measures 14 cm by 20 cm, and 160 cm^2 of picture shows. Find the width of the fr

A picture frame measures 14 cm by 20 cm, and 160 cm^2 of...

Key Concepts

Uniform Reduction of Dimensions

When a frame has a consistent width on all sides, the visible dimensions of the picture are reduced by twice the frame's width in both the length and the width. Recognizing this helps in accurately expressing the smaller, inner dimensions relative to the overall dimensions of the frame.

Quadratic Equation Solving

The process of setting up the equation for the area of the visible picture naturally leads to a quadratic equation. Solving this equation involves expanding the product, simplifying the terms, and applying methods such as factoring or the quadratic formula to find the frame's width.

Algebraic Modeling

This concept involves converting a real-world scenario into an algebraic equation by assigning a variable to the unknown quantity—in this case, the width of the frame—and expressing the visible dimensions of the picture in terms of that variable.

Transcript

00:01 All right, so the question that we're looking at today is we have a picture frame that measures 14 centimeters by 20 centimeters.

00:09 So i'm just going to get that down.

00:12 So 14 cm by 20 cm.

00:20 160 cubic centimeters of the picture shows inside the frame.

00:24 So we need to find the width of the frame.

00:27 So the first thing we're going to do is we're going to find out the actual area.

00:31 Of the whole picture frame, not just what is showing in the picture.

00:37 So we do this by doing 14 centimeters just by 20 centimeters, the simple length times width area formula.

00:46 So this is going to give us 280 centimeters.

00:54 So that was the first thing.

00:57 And then we're going to set up our formula to figure out what the the width is.

01:07 So this is going to be represented by 14 minus 2w and 20 minus 2w equals 160, which is given to us in the question.

01:23 This w equals the width.

01:25 So we're going to use this, and we're going to try and get it into a polynomial formula in order for us to be able to factor it and find our solutions a lot easier.

01:33 The first thing you can do in both of these is you can factor a two from both of these formulas.

01:39 So this one would become two outside and seven minus w.

01:44 And then another two outside here to make it 10 minus w equals 160.

01:51 No, not 116, 160.

01:57 My bad.

02:00 These are going to just become four on the outside here, seven minus w.

02:06 Just simply the distributed property, i just moved it over there, and when they're together, they can be multiplied to four.

02:13 It's w equals 160.

02:18 I don't really like that four out front.

02:20 It's just, it doesn't really do a whole lot to a question.

02:23 So i'm going to divide it out by both sides.

02:29 Cancel that out.