Chapter 13.3, Problem 64ES

The area A of a triangle is given by $A=\frac{1}{2}ab\sin \theta ,$ where a and b are the lengths of two sides and $\theta$ is the angle between these sides. Suppose that $a=5,b=10,\text{and }\theta =\pi /3.$

(a) Find the rate at which A changes with respect to a if b and $\theta$ are held constant.

(b) Find the rate at which A changes with respect to $\theta$ if a and b are held constant.

(c) Find the rate at which b changes with respect to a if A and $\theta$ are held constant.