(a) A rectangular gasoline tank can hold 50.0 kg of gasoline when full. What is the depth of the tank if it is 0.500-m wide by 0.900-m long? (b) Discuss whether this gas tank has a reasonable volume for a passenger car.

Solve the given problems by solving the appropriate differential equation.

The rate of change of the radial stress $S$ on the walls of a pipe with respect to the distance $r$ from the axis of the pipe is given by $r \frac{d S}{d r}=2(a-S)$, where $a$ is a constant. Solve for $S$ as a function of $r$.

A diffraction grating 20.0 mm wide has 6000 rulings. Light of wavelength 589 nm is incident perpendicularly on the grating. What is the second largest value of $\theta$ at which maxima appear on a distant viewing screen?

Aslan and Shelley are finding the solution for $2 \sin ^{2} \theta=\sin \theta,$ $0<\theta \leq \pi$, Here is their work.

$$\begin{aligned} 2 \sin ^{2} \theta &=\sin \theta & & \\ \frac{2 \sin ^{2} \theta}{\sin\theta} &=\frac{\sin \theta}{\sin \theta} & & \text { Step } 1 \\ 2 \sin \theta &=1 & & \text { Step } 2 \\ \sin \theta &=\frac{1}{2} & & \text { Step } 3 \\ \theta &=\frac{\pi}{6}, \frac{5 \pi}{6} & & \text { Step } 4 \end{aligned}$$

a) Identify the error that Aslan and Shelley made and explain why their solution is incorrect. b) Show a correct method to determine the solution for $2 \sin ^{2} \theta=\sin \theta, 0<\theta \leq \pi$.

One of $\sin \theta, \cos \theta, \text { and } \tan \theta$ is given. Find the other two if $\theta$ lies in the specified interval. $\cos \theta=\frac{1}{3}, \quad \theta \text { in }\left[-\frac{\pi}{2}, 0\right]$

If

$$f ( x ) = 1 + x ^ { 2 }$$

and

$$g ( x ) = \sqrt { x - 1 }$$

, find the following. (a)

$$f \circ g$$

(b)

$$g \circ f$$

(c)

$$( f \circ g ) ( 2 )$$

(d)

$$( f \circ f ) ( 2 )$$

(e)

$$f \circ g \circ f$$

(f)

$$g \circ f \circ g$$

In this exercise, prove the following given identity:

$$\tan \theta+\frac{1}{\csc \theta}=\frac{1+\cos \theta}{\cot \theta}$$

Given that cos $\theta=-\frac{3}{5}$ and $\frac{\pi}{2}<\theta<\pi$, find each of the following. (a) $\cos 2 \theta$, (b) $\sin 2 \theta$, (c) $\tan 2 \theta$, (d) $\cos \frac{1}{2} \theta$, (e) $\tan \frac{1}{2} \theta$.

A bookmark is shaped like a rectangle with a semicircle attached at both ends. The rectangle is 12 cm long and 4 cm wide. The diameter of each semicircle is the width of the rectangle. What is the area of the bookmark? Use 3.14 for $\pi$.

How many ways can president and vice president be determined in a club with twelve members?

If $f(x)=x+5$ and $g(x)=x^2-3$, find the following.

$$g(g(x))$$

Find

$$\sin 2 \theta , \cos 2 \theta , \tan 2 \theta , \sin \frac { \theta } { 2 } , \cos \frac { \theta } { 2 } , \text { and } \tan \frac { \theta } { 2 }$$

for the set of conditions.

$$\sin \theta = \frac { 2 } { 5 } \text { and } 0 ^ { \circ } < \theta < 90 ^ { \circ }$$

Find the second derivative of the function.

$$f(t)=t^2e^{-2t}$$

(a) Use graphs of the following to find the number of zeros the function has on the interval $0 \leq \theta \leq 4 \pi$.

(i) $f(\theta)=2 \sin \theta+1$

(ii) $g(\theta)=2 \sin \theta+3$

(iii) $h(\theta)=-2 \sin \theta+2$

(b) How must $A$ and $k$ be related so that the function $f(\theta)=A \sin \theta+k$ has no zeros on this interval?

(c) How must $A$ and $k$ be related so that the function $f(\theta)=A \sin \theta+k$ has exactly two zeros on the interval $0 \leq \theta \leq 4 \pi$ ?