34. In the figure above, $P Q$ represents a 40-foot ladder with end $P$ against a vertical wall and end $Q$ on level ground. If the ladder is slipping down the wall, what is the distance $R Q$ at the instant when $Q$ is moving along the ground $\frac { 3 } { 4 }$ as fast as $P$ is moving down the wall?
(A) $\frac { 6 } { 5 } \sqrt { 10 }$
(B) $\frac { 8 } { 5 } \sqrt { 10 }$
(C) $\frac { 80 } { \sqrt { 7 } }$
(D) 24
(E) 32

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

(Calculus) Point $\mathrm{A}$ is on circle $\mathrm{O}$ with radius 1. As shown in the figure, for a positive angle $\theta$, two points $\mathrm{B}$ and $\mathrm{C}$ on circle $\mathrm{O}$ are chosen such that $\angle \mathrm{BAC} = \theta$ and $\overline{\mathrm{AB}} = \overline{\mathrm{AC}}$. Let $r(\theta)$ denote the radius of the inscribed circle of triangle $\mathrm{ABC}$. If $\lim_{\theta \rightarrow \pi - 0} \frac{r(\theta)}{(\pi - \theta)^2} = \frac{q}{p}$, find the value of $p^2 + q^2$. (Given: $p$ and $q$ are coprime natural numbers.) [4 points]

For a natural number $n$, let P be the point with coordinates $( 0,2 n + 1 )$, and let Q be the point on the graph of the function $f ( x ) = n x ^ { 2 }$ with $y$-coordinate 1 in the first quadrant. For the point $\mathrm { R } ( 0,1 )$, let $S _ { n }$ be the area of triangle PRQ and $l _ { n }$ be the length of line segment PQ. What is the value of $\lim _ { n \rightarrow \infty } \frac { S _ { n } ^ { 2 } } { l _ { n } }$? [4 points]
(1) $\frac { 3 } { 2 }$
(2) $\frac { 5 } { 4 }$
(3) 1
(4) $\frac { 3 } { 4 }$
(5) $\frac { 1 } { 2 }$

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) $\frac { 5 } { 8 }$

As shown in the figure, there is a rhombus ABCD with side length 1. Let E be the foot of the perpendicular from point C to the extension of segment AB, let F be the foot of the perpendicular from point E to segment AC, and let G be the intersection of segment EF and segment BC. If $\angle \mathrm { DAB } = \theta$, let the area of triangle CFG be $S ( \theta )$.
What is the value of $\lim _ { \theta \rightarrow 0 + } \frac { S ( \theta ) } { \theta ^ { 5 } }$? (Here, $0 < \theta < \frac { \pi } { 2 }$) [4 points]
(1) $\frac { 1 } { 24 }$
(2) $\frac { 1 } { 20 }$
(3) $\frac { 1 } { 16 }$
(4) $\frac { 1 } { 12 }$
(5) $\frac { 1 } { 8 }$

112- Point $M(x,2)$ lies on the curve $y=2$. It is a variable point. The line segment connecting point $M$ to the origin, makes an angle $\alpha$ with the positive $x$-axis. The rate of change of $\alpha(x)$ with respect to $x$, at the moment $x=4$, is which of the following?
(1) $-0/2$ (2) $-0/1$ (3) $0/05$ (4) $0/15$