Elliptical concrete pipes have a greater capacity for shallow flow than circular pipes. For this reason, elliptical pipes are often used in storm drains, culverts, and sewers. An engineer wants to design an elliptical concrete pipe with a maximum horizontal opening of 4 ft and a maximum vertical opening of 3 ft. a. Write an equation for an elliptical cross section of the pipe. For convenience, place the coordinate system with (0,0) at the center of the pipe. b. To construct the pipe, the engineer needs to know the location of the foci. How far from the center are the foci?

Acetic acid, $\mathrm{HC}_2 \mathrm{H}_3 \mathrm{O}_2$, is responsible for the sour taste of vinegar. Combustion of acetic acid gives off $14.52 \mathrm{~kJ} / \mathrm{g}$ of heat. When $15.00 \mathrm{~g}$ of acetic acid is burned in a bomb calorimeter (heat capacity $=2.166 \mathrm{~kJ} /{ }^{\circ} \mathrm{C}$ ) containing $0.800 \mathrm{~kg}$ of water, the final temperature in the calorimeter is $64.14^{\circ} \mathrm{C}$. What was the initial temperature of the calorimeter?

A concave spherical mirror has a radius of curvature of $6 \mathrm{~cm}$. A point object is on the axis $9 \mathrm{~cm}$ from the mirror. Construct a precise ray diagram showing rays from the object that make angles of $5^{\circ}, 10^{\circ}, 30^{\circ}$, and $60^{\circ}$ with the axis, which strike the mirror and are reflected back across the axis. (Use a compass to draw the mirror, and use a protractor to measure the angles needed to find the reflected rays.) What is the spread $\delta x$ of the points where these rays cross the axis?

In a study of the effects of food deprivation on hunger,$^{9}$ an insect was fed until its appetite was completely satisfied. Then it was deprived of food for t hours (the deprivation period. At the end of this period, the insect was re-fed until its appetite was again completely satisfied. The weight H (in grams) of the food that was consumed at this time was statistically found to be a function of t, where $H=1.00\left[1-e^{-(0.0464 t+0.0670)}\right].$ Here H is a measure of hunger. Show that H is increasing with respect to t and is concave down.

A laser beam is incident at an angle of $30.0 ^ { \circ }$ to the vertical onto a solution of corn syrup in water. If the beam is refracted to $19.24 ^ { \circ }$ to the vertical, Find speed in the solution.

A double concave lens of glass with $n=1.53$ has surfaces of $5 \mathrm{D}$ (diopters) and $8 \mathrm{D}$, respectively. The lens is used in air and has an axial thickness of $3 \mathrm{~cm}$. a. Determine the position of its focal and principal planes. b. Also find the position of the image, relative to the lens center, corresponding to an object at $30 \mathrm{~cm}$ in front of the first lens vertex. c. Calculate the paraxial image distance assuming the thinlens approximation. What is the percent error involved?

Larry is renting an apartment that will cost r dollars per month. He must pay a D dollars application fee and C dollars for a credit application. His security deposit is two month's rent and he must also pay the last month's rent upon signing the lease. His broker charges 7% of the total year's rent as the fee for finding the apartment. Write an algebraic expression that represents the total cost of signing the lease.

A lens has one convex surface of radius $6.00 \mathrm{~cm}$ and one concave surface of radius $10.0 \mathrm{~cm}$. When an object is placed $35.0 \mathrm{~cm}$ from the lens, a real image is formed $50 \mathrm{~cm}$ from the lens.
(a) What is the focal length of the lens?
(b) What is the index of refraction of the lens?

In the following table the length $L$, in inches, of a flying animal and its maximum speed $F$, in feet per second, when it flies. ${ }^{27}$ (For comparison, 10 feet per second is about $6.8$ miles per hour.)

$$\begin{array}{|l|c|c|} \hline\text{Animal} & \text{Length L}& \begin{array}{c} \text { Flying speed } \\ \text { F } \end{array} \\ \hline \text { Fruit fly } & 0.08 & 6.2 \\ \hline \text { Horse fly } & 0.51 & 21.7 \\ \hline \begin{array}{l} \text { Ruby-throated } \\ \text { hummingbird } \end{array} & 3.2 & 36.7 \\ \hline \text { Willow warbler } & 4.3 & 39.4 \\ \hline \text { Flying fish } & 13 & 51.2 \\ \hline \text { Bewick's swan } & 47 & 61.7 \\ \hline \text { White pelican } & 62 & 74.8 \\ \hline \end{array}$$

a. Judging on the basis of this table, is it generally true that larger animals fly faster? b. Find a formula that models $F$ as a power function of $L$. c. Draw the graph of the function in part b. d. Is the graph you found in part c concave up or concave down? Describe in practical terms what your answer means. e. How much quicker would you anticipate a bird to fly if it were 10 times longer than another? (Use the homogeneity property of power functions.)

An object located $100 \mathrm{~cm}$ from a concave mirror forms a real image $75 \mathrm{~cm}$ from the mirror. The mirror is then turned around so that its convex side faces the object. The mirror is moved so that the image is now $35 \mathrm{~cm}$ behind the mirror. How far was the mirror moved? Was it moved toward the object or away from the object?

The economic advisor of a large tire store proposes the demand function D(p)=

$$\frac { 1800 } { p - 40 }$$

, where D(p) is the number of tires of one brand and size that can be sold in one day at a price p. For what prices is the demand elastic? Inelastic?

We use the vertical line test to determine whether a graph is of a function. What do you know about the definition of a function, explain the vertical line test.

In the following exercise, sketch the graph of the plane curve given by the vector-valued function and, at the point on the curve determined by $r\left(t_0\right)$, sketch the vectors $T$ and $N$. Note that $\mathbf{N}$ points toward the concave side of the curve.

Function: $\mathbf{r}(t)=t \mathbf{i}+\frac{1}{t} \mathbf{j}$, Time: $t_0=2$

Use properties of determinants to find the value of each determinant if it is known that

$$\left|\begin{array}{lll}{x} & {y} & {z} \\ {u} & {v} & {w} \\ {1} & {2} & {3}\end{array}\right|=4$$

.

$$\left|\begin{array}{rrr}{x} & {y} & {z} \\ {-3} & {-6} & {-9} \\ {u} & {v} & {w}\end{array}\right|$$