Two sides of a triangle have lengths 6 cm and 8 cm, and the angle bet…

The correct answer is Option (1) → 24 cm²

Let θ be the angle between the two sides, \(0 \lt \theta \lt \pi\). The area is

    \( \displaystyle A = \frac{1}{2} \times 6 \times 8 \times \sin\theta = 24\sin\theta \)

Setting the derivative equal to zero:

    \( \displaystyle \frac{dA}{d\theta} = 24\cos\theta = 0,\ \theta = \frac{\pi}{2} \)

Since \(\frac{d^{2}A}{d\theta^{2}} = -24\sin\theta = -24 \lt 0\) there, this is a maximum:

    \( \displaystyle A = 24\sin\frac{\pi}{2} = 24 \) cm²

The area is largest when the two sides are at right angles.