Optimizing a Height Function Using Trigonometric Constraints

00:00 3 marks

A civil engineer is analyzing a structure whose height profile is modeled by the function \(f(\theta) = 5\sin(\theta) + 12\cos(\theta)\). The design requires determining the optimal angle \(\theta\) (where \(0 \leq \theta \leq \frac{\pi}{2}\)) that maximizes the height for structural efficiency.

(a) Calculate the value of \(\theta\) (in radians) that maximizes \(f(\theta)\).

(b) Determine the maximum height achieved by the function.

Hint Hint

Hint

Differentiate \(f(\theta)\) with respect to \(\theta\) and set the derivative equal to zero to find critical points. Consider the domain of \(\theta\) to ensure the maximum is within the given range.

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Answer Answer

Complete Answer

The maximum height is 13, achieved when \(\theta = \arctan\left(\frac{5}{12}\right)\).

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