Express \(7 \cos \theta + 24 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
As \(\beta\) varies, the greatest possible value of
$$\frac { 150 } { 7 \cos \frac { 1 } { 2 } \beta + 24 \sin \frac { 1 } { 2 } \beta + 50 }$$
is denoted by \(V\).
Find the value of \(V\) and determine the smallest positive value of \(\beta\) (in degrees) for which the value of \(V\) occurs.
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