Problem 8E

8. The area of a triangle with sides of lengths $a$ and $b$ and contained angle $\theta$ is

$$

A=\frac{1}{2} a b \sin \theta

$$

(a) If $a=2 \mathrm{~cm}, b=3 \mathrm{~cm}$, and $\theta$ increases at a rate of $0.2 \mathrm{rad} / \mathrm{min}$, how fast is the area increasing when $\theta=\pi / 3 ?$

(b) If $a=2 \mathrm{~cm}, b$ increases at a rate of $1.5 \mathrm{~cm} / \mathrm{min}$, and $\theta$ increases at a rate of $0.2 \mathrm{rad} / \mathrm{min}$, how fast is the area increasing when $b=3 \mathrm{~cm}$ and $\theta=\pi / 3$ ?

(c) If $a$ increases at a rate of $2.5 \mathrm{~cm} / \mathrm{min}, b$ increases at a rate of $1.5 \mathrm{~cm} / \mathrm{min}$, and $\theta$ increases at a rate of $0.2 \mathrm{rad} / \mathrm{min}$, how fast is the area increasing when $a=2 \mathrm{~cm}, b=3 \mathrm{~cm}$, and $\theta=\pi / 3$ ?

Step-by-Step Solution