calculus 1 Question Rate of Change

Question

Let θ be an angle that is not a right angle in a right triangle. If the ratio of the length of the adjacent side to the opposite side is increasing at a rate of 1/3 s−1 when the lengths of the two sides are respectively 4 and 5 ft, what is the rate of change of θ with respect to time at that time.

Answer 1

cotθ = y/x if x is opposite leg and y is adjaicent leg: d/dt(y/x) = - csc²θ(dθ/dt) = 1/3

csc²θ = (x² + y²)/y² = (5² + 4²)/4² = 41/16. So, dθ/dt = - (1/3)·16/41 = - 16/123 s^-1

Answer 2

The ratio of the adjacent to the opposite side = cot(θ)

d cot(θ)/dt = -csc²(θ) dθ/dt = 1/3

dθ/dt = -1/(3csc²(θ)) = -sin²(θ)/3

When adjacent = 4 and opposite =5, hypotenuse = √41

sin(θ) = 5/√41 and sin²(θ) = 25/41

dθ/dt = -(1/3)(25/41) = -25/123 radians/s