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Problem Bank

[Maximum mark: 19]

Anne is designing a gift box in the shape of a right triangular prism. In order for the box to be shipped, the longest side of the front face of the box must be 1m.

Let the other side lengths of the box be p and q. Anne's company also sells wrapping paper. Hence, in order to maximize profits, Anne wants to maximize the surface area of the box.

1. Find the perimeter, P of the front (triangular face) of the box in terms of p and q.
  [1]
2. Show that P=1+p+√1−p2.
  [2]
Find dpdP.
[4]
Find the value of p maximizing perimeter.
[2]
Give an expression for the area of the triangle in terms of p.
[3]
Show that the value of p maximizing the perimeter also maximizes the area.
[5]
Hence find the value of p maximizing the surface area of the box.
[2]

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