9.
Figure 3
$$( x + 1 ) ^ { 2 }$$
Figure 3 shows a triangle \(P Q R\). The size of angle \(Q P R\) is \(30 ^ { \circ }\), the length of \(P Q\) is \(( x + 1 )\) and the length of \(P R\) is \(( 4 - x ) ^ { 2 }\), where \(X \in \Re\).
- Show that the area \(A\) of the triangle is given by \(A = \frac { 1 } { 4 } \left( x ^ { 3 } - 7 x ^ { 2 } + 8 x + 16 \right)\)
- Use calculus to prove that the area of \(\triangle P Q R\) is a maximum when \(x = \frac { 2 } { 3 }\).
Explain clearly how you know that this value of \(x\) gives the maximum area.
- Find the maximum area of \(\triangle P Q R\).
- Find the length of \(Q R\) when the area of \(\triangle P Q R\) is a maximum.
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