The post office has requirements on the sizes of boxes that are mailed as parcels. The measurement “around” the box must be no longer than 84” with the height of the box no more than 30”. Assuming the height is fixed at 30”, generate a quadratic function to model the area of the top of the box, using 84” as the perimeter.

Define the length (l) and the width (w) as the dimensions of the box top/bottom. A table can be generated from knowing that one width and one length of the top will be half the perimeter or 42”...

The perimeter: 2l + 2w = 84. The table led us to see that the length is 42-w, so we can now write an equation for the area of the top: A = w(42 - w) = 42w - w².

For this description, you can generate an exact equation because we are given enough information to write the particular equation and to build a table of exact values. Now, you can graph the equation using our calculator and observe the relationship between the lengths of the sides of this lid and the area of the lid.

Notice the three different looking equations I entered into the Y = menu on my calculator.

Are they the same function?

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Yes, just different algebraic forms.

Looking at the graph, notice I have Traced over to the point on the graph when x = 21.

What is the meaning of the y value of 441 here?

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The y value of 441 is the area when the width is 21 (which means the length would also be 21 as it is equal to 42-width).