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Students explore the relationship between length and area of a rectangle.

\n

The perimeter of the rectangle is randomised. Students are given 11 different lengths, and asked to compute rectangle width and area for each. They are then asked to graph the function, identify it as a parabola, and estimate the maximum value.

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You have $\\var{p}$ metres of fencing with which to build a rectangular pen.

\n

You are going to explore the amount of area within the pen that you can achieve with different side lengths.

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$Perimeter = 2 \\times length + 2 \\times width$.

\n

So $width = (perimeter - 2 \\times length)\\div 2$

\n

$Area = length \\times width$

\n

The graph of area plotted against length looks like this:

\n

\n

This is a parabola.

\n

The maximum value from the table is {maxareafromtable}. The maximum area from the graph occurs when the length is {xmax} and is equal to {ymax}.

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the perimeter

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first length

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the shortest length

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Complete the following table.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
length{lengths[0]}{lengths[1]}{lengths[2]}{lengths[3]}{lengths[4]}{lengths[5]}{lengths[6]}{lengths[7]}{lengths[8]}{lengths[9]}{lengths[10]}
width[[0]][[1]][[2]][[3]][[4]][[5]][[6]][[7]][[8]][[9]][[10]]
area[[11]][[12]][[13]][[14]][[15]][[16]][[17]][[18]][[19]][[20]][[21]]
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Now use the information in the table to plot a graph of area against length.

\n

At the end, click the \"Reveal answers\" button to see a sample graph.

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What form does the graph have?

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Estimate the maximum area from your table or graph.

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