two sides of a triangle have lengths 5 m and 10 m. the angle between them is increasing at a rate of 0.06…

Explanation:

Step1: Recall area formula for triangle

The area formula for a triangle with two - side lengths (a) and (b) and the included angle (\theta) is (A=\frac{1}{2}ab\sin\theta). Here (a = 5), (b = 10), so (A=\frac{1}{2}(5)(10)\sin\theta=25\sin\theta).

Step2: Differentiate with respect to time (t)

Using the chain - rule (\frac{dA}{dt}=\frac{dA}{d\theta}\cdot\frac{d\theta}{dt}). Since (A = 25\sin\theta), then (\frac{dA}{d\theta}=25\cos\theta). We are given that (\frac{d\theta}{dt}=0.06) rad/s.

Step3: Substitute the value of (\theta)

We know that (\theta = 60^{\circ}=\frac{\pi}{3}) radians, and (\cos\frac{\pi}{3}=\frac{1}{2}). Then (\frac{dA}{dt}=25\cos\theta\cdot\frac{d\theta}{dt}). Substitute (\theta=\frac{\pi}{3}) and (\frac{d\theta}{dt}=0.06) into the equation: (\frac{dA}{dt}=25\times\frac{1}{2}\times0.06).

Step4: Calculate the result

(\frac{dA}{dt}=25\times\frac{1}{2}\times0.06 = 0.75) (m^{2}/s).

Answer:

0.75 (m^{2}/s)