A triangular channel with side slopes having a ratio of $1: 1.5$ carries $0.68 \mathrm{ft}^3 / \mathrm{s}$ of water and is made from clean, excavated earth. Use $y=0.25 \mathrm{ft}$ in (d).

a. Compute the critical depth. b. Compute the minimum specific energy. c. Plot the specific energy curve. d. Determine the specific energy for the given depth and the alternate depth for this energy. e. Determine the velocity of flow and the Froude number for each depth in (d). f. Compute the required slopes of the channel if the depths from (d) are to be normal depths for the given flow rate.

Suppose that $f$ is differentiable at $\left(x_0, y_0\right)$ and $\nabla f\left(x_0, y_0\right) \neq$ $\mathbf{0}$. Calculate the rate of change of $f$ in the direction of

$$\frac{\partial f}{\partial y}\left(x_0, y_0\right) \mathbf{i}-\frac{\partial f}{\partial x}\left(x_0, y_0\right) \mathbf{j} \text {. }$$

Give a geometric interpretation to your answer.

Ms. Augello is making tie dye shirts with her students. Each gallon of hot water needs

$$\frac{2}{3}$$

cup of tie dye. If Ms. Augello has

$$5\frac{1}{4}$$

cups of tie dye, how many batches of solution will she be able to make?

Show that $u(x, t)=t^{\alpha} F(\eta)$, where $\eta=x^{2} / t$ is a solution of the partial differential equation $\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}$ if F satisfies $4 \eta \frac{\mathrm{d}^{2} F}{\mathrm{d} \eta^{2}}+(2+\eta) \frac{\mathrm{d} F}{\mathrm{d} \eta}-\alpha F=0$ Find non-zero values of α and κ for which $F=\mathrm{e}^{\kappa \eta}$ is a solution.

The Fitzgeralds are installing inground pool in their back yard. They can either install the base model offered by the pool company or add any of the following options: automatic pool cleaner, solar cover, waterfall, hot tub, fountain, slide, diving board. How many different variations of the pool are possible?

A sink can be filled by the hot-water tap alone in $4$ minutes more than it takes the cold-water tap alone. If both taps are open, it takes $4$ minutes, $48$ seconds to fill an empty sink. How long does it take each tap individually to fill the sink?

The monthly mortgage payment in dollars, $P$, for a house is a function of three variables

$$P=f(A, r, N),$$

where $A$ is the amount borrowed in dollars, $r$ is the interest rate, and $N$ is the number of years before the mortgage is paid off. $f(92000,14,30)=1090.08$. $\left.\frac{\partial P}{\partial r}\right|_{(92000,14,30)}=72.82$. Would you expect $\partial P / \partial N$ to be positive or negative? Why?

Determine the partial derivative with respect to x and the partial derivative with respect to y at x=3 and y=2 for $f(x, y)=7 x^2 y^5+2 y+5 x^3+1$.

Sketch a magnified view (showing atoms/molecules) of each of the following and explain: a homogeneous mixture of an element and a compound

Spam filters try to sort your e-mails, deciding which are real messages and which are unwanted. One method used is a point system. The filter reads each incoming e-mail and assigns points to the sender, the subject, key words in the message, and so on. The higher the point total, the more likely it is that the message is unwanted. The filter has a cutoff value for the point total; any message rated lower than that cutoff passes through to your inbox, and the rest, suspected to be spam, are diverted to the junk mailbox. We can think of the filter's decision as a hypothesis test. The null hypothesis is that the e-mail is a real message and should go to your inbox. A higher point total provides evidence that the message may be spam; when there's sufficient evidence, the filter rejects the null, classifying the message as junk. This usually works pretty well, but, of course, sometimes the filter makes a mistake. Which kind of error is it when a real message gets classified as junk?

Why might some rear-view mirrors in a car state, "Objects are closer than they appear"?

Each family in a neighborhood is contributing $\$ 20$ worth of food to the neighborhood picnic. The Harlin family is bringing 12 packnges of buns. The hamburger buns cost $\$ 2.00$ per package. The hot-dog buns cost $\$ 1.50$ per package. How many packages of each type of bun did they buy?

Write an equation of the line with the given characteristics.

points: $(5,0),(-3,2)$

Why do we study the behavioral characteristics of monozygotic twins?