A=(1)/(2) x^2 sinθ(d A)/(dt)=x sinθ(dx)/(dt)+(x^2)/(2) cosθ(d θ)/(dt) =12 ×(1)/(√(2)) ×(1)/(12

step 1 This triangle is isisclos and we have to find rate of change of area of strangles and this is tw

A=(1)/(2) x^2 sinθ(d A)/(dt)=x sinθ(dx)/(dt)+(x^2)/(2)...

$\mathrm{A}=\frac{1}{2} \mathrm{x}^{2} \sin \theta$ $\frac{\mathrm{d} \mathrm{A}}{\mathrm{dt}}=\mathrm{x} \sin \theta \frac{\mathrm{dx}}{\mathrm{dt}}+\frac{\mathrm{x}^{2}}{2} \cos \theta \frac{\mathrm{d} \theta}{\mathrm{dt}}$ $\quad=12 \times \frac{1}{\sqrt{2}} \times \frac{1}{12}+\frac{144}{2} \times \frac{1}{\sqrt{2}} \times \frac{\pi}{180}$ $=2^{1 / 2}\left(\frac{1}{2}+\frac{\pi}{5}\right)$

$\mathrm{A}=\frac{1}{2} \mathrm{x}^{2} \sin \theta$
$\frac{\mathrm{d} \mathrm{A}}{\mathrm{dt}}=\mathrm{x} \sin \theta \frac{\mathrm{dx}}{\mathrm{dt}}+\frac{\mathrm{x}^{2}}{2} \cos \theta \frac{\mathrm{d} \theta}{\mathrm{dt}}$
$\quad=12 \times \frac{1}{\sqrt{2}} \times \frac{1}{12}+\frac{144}{2} \times \frac{1}{\sqrt{2}} \times \frac{\pi}{180}$
$=2^{1 / 2}\left(\frac{1}{2}+\frac{\pi}{5}\right)$

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Key Concepts

Differentiation with Respect to Time

Differentiation with respect to time involves taking the derivative of a function in which the variables are functions of time. This concept is crucial when analyzing dynamic systems and rates of change, as demonstrated in the calculation of dA/dt in the problem.

Trigonometric Derivatives

Trigonometric derivatives refer to the process of finding the derivative of trigonometric functions. In the given problem, the derivative of sin(theta) with respect to time is computed as cos(theta) dtheta/dt, which is essential for correctly determining the rate of change in area.

Product Rule

The product rule is a fundamental technique in calculus used to differentiate expressions where two or more functions are multiplied together. In this problem, the area A is expressed as a product of functions involving x and sin(theta), necessitating the use of the product rule to find the time derivative of A.

Chain Rule

The chain rule is a method for differentiating composite functions, particularly when one variable depends on another. Here, both x and theta are functions of time, so the chain rule is applied to correctly account for the time dependence of each variable when differentiating the expression for A.

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