Question

(1 point) Let @ (in radians) be an acute angle in a right triangle and let x and y, respectively, be the lengths of the sides adjacent to and opposite e. Suppose also that x and y vary with time. At a certain instant x = 8 units and is increasing at 4 unit/s, while y = 5 and is decreasing at į units/s. How fast is a changing at that instant? Answer:

Answer 1

We can draw triangle

8. ២/Y 3 X

we are given

$x=8,y=5$

$\frac{dx}{dt}=4,\frac{dy}{dt}=-\frac{1}{6}$

we can use sine formula

sin(0) y 89

we can find derivative with respect to t

$\frac{d\theta}{dt}cos(\theta)=\frac{\frac{dy}{dt}}{\sqrt{89}}$

now, we can plug cos value

$\frac{d\theta}{dt}(\frac{8}{\sqrt{89}})=\frac{\frac{dy}{dt}}{\sqrt{89}}$

$\frac{d\theta}{dt}(\frac{8}{\sqrt{89}})=\frac{-\frac{1}{6}}{\sqrt{89}}$

$\frac{d\theta}{dt}=\frac{\frac{-\frac{1}{6}}{\sqrt{89}}}{(\frac{8}{\sqrt{89}})}$

so, we get

$\frac{d\theta}{dt}=-\frac{1}{48} rad/sec$...........Answer