Question

Consider the given right triangle with legs of length x cm. and y cm. and angle θ radians. If x is decreasing at the rate of 3 cm/min. and y is increasing at the rate of 4 cm/min., at what rate is angle θ changing when x=5 cm. and y=2 cm. ?

Answer 1

$y=xtan\theta$

$\theta=tan^{-1}(y/x)$

$\frac{d\theta}{dt}=\frac{1}{1+(y/x)^2}\cdot \frac{y'x-x'y}{x^2}=\frac{1}{1+(2/5)^2}\cdot \frac{4\cdot 5-3\cdot 2}{5^2}=\frac{14}{29}$