Question10. A triangle with base $b$ and height $h$ is expanding such that its area is increasing at the rate $4 \mathrm{~m}^{2} / \mathrm{s}$. (A) $\frac{d A}{d t}=4$ B) $\frac{d b}{d t}=4$ C) $\frac{d h}{d t}=4$ D) $\frac{d t}{d A}=4$ $\frac{d A}{d t}=4 \mathrm{~m}^{2} / \mathrm{s}$
11. A spherical rock erodes over time due to winds and rain. The radius of the rock is changing at a rate of $3 \mathrm{in} /$ year. Find the rate that the volume of the rock is changing when the surface area (A) of the rock is $400 \mathrm{in}^{2}$. A) $\frac{d A}{d t}=3$ B) $\frac{d r}{d t}=3$ (C) $\frac{d V}{d t}=-3$ D) $\frac{d r}{d t}=-3$
Directions: Solve the following problems. Show all work clearly and neatly. $$(3 r+1)^{1 / 2}$$
12. Let $s$ and $r$ be variables such that $2 s=\sqrt{3 r+1}$. Find $\frac{d r}{d t}$ when $\frac{d s}{d t}=5$ and $s=2$. $$\begin{array}{ll} 2 \frac{d s}{d t}=\frac{1}{2}(3 r+1)^{-\frac{1}{2}} \cdot 3 \frac{d r}{d t} & (2 \cdot 2)^{2}=\sqrt{3 r+1}^{2} \\ 2 \cdot 5=\frac{1}{2 \sqrt{3 r+1}} \cdot 3 \frac{d r}{d t} & 16=3 r+1 \\ 20 \sqrt{3 r+1}=3 \frac{d r}{d t} & 15=3 r \\ 20 r \end{array}$$ $$\begin{array}{l} \sqrt{3 r+1}=3 \frac{d r}{d t} \\ \frac{20 \sqrt{3 r+1}}{3}=\frac{d r}{d t}=\frac{20 \sqrt{16}}{3}=\frac{20 \cdot 4}{3}=\frac{80}{3} d x \end{array}$$
13. A particle is moving along the graph of $x y^{2}=36$ where $\frac{d y}{d t}=-2$. Find $\frac{d x}{d t}$ when $y=-3$. $$\begin{array}{ll} x(-3)^{2}=36 & \frac{d x}{d t} \cdot 2 y \frac{d y}{d t}=0 \\ x(9)=36 & \frac{d x}{d t} \cdot 2 \cdot-3 \cdot-2 \\ x=4 & \frac{d x}{d t} \cdot 12=0 \end{array}$$
14. The radius of a circle is decreasing at a rate of 3 inches per second. At what rate is the area of the circle changing when the circumference is $16 \pi$ inches?