Homework SourseLet theta in radians be an acute angle in a right triangle a
Let theta (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 to . Suppose also that x and y varies with time?
a.Give the equation that relates the rates of the quantities.
b.At a certain instant, x=2 units and is increasing at 1 unit/sec, while y=2 units
and is decreasing at (1/4) unit/sec. How fast is theta at that instant?
a.Give the equation that relates the rates of the quantities.
b.At a certain instant, x=2 units and is increasing at 1 unit/sec, while y=2 units
and is decreasing at (1/4) unit/sec. How fast is theta at that instant?
Solution
a)tan(theta(t))=y(t)/x(t) differentiate sec^2(theta(t))*theta\'(t)= y\'(t)x(t)-y(t)x\'(t)/x^2(t) thus theta\'(t)= (y\'(t)x(t)-y(t)x\'(t)) divided by (x^2(t)*sec^2(?(t)) b) tan(theta(t))=y(t)/x(t) =2/2=1 so theta=pi/4 sec^2(pi/4)=1/cos^2(pi/4) =1/(1/square root 2)^2=2 plug in theta\'(t) =(y\'(t)x(t)-y(t)x\'(t)) divided by (x^2(t)*sec^2(?(t)) =(-0.25*2-2*1)/(2^2*2) =(-2-0.5)/(8) =-2.5/8=-0.3125 rad/s
