Wire split and minimum combined area

Wire split and minimum combined area

Secondary 2
8 marks
TGM Original Questions

Optional extension

An \(80\) m wire is split. One part forms a rectangle \(6x\) m by \(2x\) m; the remainder forms a square.

  1. Show the total area is \(A=28x^2-160x+400\).[2]
    1. complete the missing table values \(a,b\)[2]
    2. plot for \(1\le x\le4.5\)[2]
    3. estimate the \(x\) giving minimum area.[2]
\(x\)\(1\)\(1.5\)\(2\)\(2.5\)\(3\)\(3.5\)\(4\)\(4.5\)
\(A\)\(268\)\(223\)\(192\)\(a\)\(172\)\(183\)\(b\)\(247\)

For graph readings, give values to the nearest 0.1 unit where an exact value cannot be read from the grid. Use the labelled axes and grid provided.

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Answer:(a) \(A=28x^2-160x+400\) (shown). (b)(i) \(a=175,b=208\) (iii) \(x\approx2.9\) m; minimum area \(\approx171.43\text{ m}^2\).

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