Ship $A$ is traveling due west toward Lighthouse Rock at a speed of 15 kilometers per hour ($\mathrm{km/hr}$). Ship $B$ is traveling due north away from Lighthouse Rock at a speed of $10\mathrm{~km/hr}$. Let $x$ be the distance between Ship $A$ and Lighthouse Rock at time $t$, and let $y$ be the distance between Ship $B$ and Lighthouse Rock at time $t$, as shown in the figure above.
(a) Find the distance, in kilometers, between Ship $A$ and Ship $B$ when $x = 4\mathrm{~km}$ and $y = 3\mathrm{~km}$.
(b) Find the rate of change, in $\mathrm{km/hr}$, of the distance between the two ships when $x = 4\mathrm{~km}$ and $y = 3\mathrm{~km}$.
(c) Let $\theta$ be the angle shown in the figure. Find the rate of change of $\theta$, in radians per hour, when $x = 4\mathrm{~km}$ and $y = 3\mathrm{~km}$.

[10 points] A spider starts at the origin and runs in the first quadrant along the graph of $y = x^{3}$ at the constant speed of 10 unit/second. The speed is measured along the length of the curve $y = x^{3}$. The formula for the curve length along the graph of $y = f(x)$ from $x = a$ to $x = b$ is $\ell = \int_{a}^{b} \sqrt{1 + f'(x)^{2}}\, dx$. As the spider runs, it spins out a thread that is always maintained in a straight line connecting the spider with the origin. What is the rate in unit/second at which the thread is elongating when the spider is at $\left(\frac{1}{2}, \frac{1}{8}\right)$?
You should use the following names for variables. At any given time $t$, the spider is at the point $\left(u, u^{3}\right)$, the length of the thread joining it to the origin in a straight line is $s$ and the curve length along $y = x^{3}$ from the origin till $\left(u, u^{3}\right)$ is $\ell$. You are asked to find $\frac{ds}{dt}$ when $u = \frac{1}{2}$. (Do not try to evaluate the integral for $\ell$: it is unnecessary and any attempt to do so will not get any credit because a closed formula in terms of basic functions does not exist.)

The distance between point O and point E is 40 m. As shown in the figure on the right, person A departs from point O and runs along the half-line OS perpendicular to segment OE at a constant speed of 3 m/s, and person B departs from point E 10 seconds after person A starts and runs along the half-line EN perpendicular to segment OE at a constant speed of 4 m/s. The angle formed by the intersection of the segment connecting the positions of persons A and B with segment OE is $\theta$ (in radians). What is the rate of change of $\theta$ at the moment 20 seconds after person A departs? [4 points]
(1) $\frac { 21 } { 290 }$ radians/second
(2) $\frac { 13 } { 290 }$ radians/second
(3) $\frac { 7 } { 290 }$ radians/second
(4) $\frac { 3 } { 290 }$ radians/second
(5) $\frac { 1 } { 290 }$ radians/second

As shown in the figure, there is a sector OAB with radius 1 and central angle $\frac { \pi } { 2 }$. Let H be the foot of the perpendicular from point P on arc AB to line segment OA, and let Q be the intersection of line segment PH and line segment AB. When $\angle \mathrm { POH } = \theta$, let $S ( \theta )$ be the area of triangle AQH. What is the value of $\lim _ { \theta \rightarrow 0 + } \frac { S ( \theta ) } { \theta ^ { 4 } }$? (Here, $0 < \theta < \frac { \pi } { 2 }$) [4 points]
(1) $\frac { 1 } { 8 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 1 } { 2 }$
(5) $\frac { 5 } { 8 }$