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?

A 4-ft-diameter finished-concrete sewer pipe is half full of water. (a) In the spirit, estimate the speed of propagation of a small-amplitude wave propagating along the channel. (b) Estimate the Froude number if the water is flowing at $14,000 \mathrm{gal} / \mathrm{min}$.

Find the indicated partial sum of the series. $\sum_{i=1}^{\infty} \frac{7}{5^{i}}$ Fourth partial sum

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.

Draw and Label the bones and features indicated on the anterior view of the Right knee , using the terms provided
Tibia

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?

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.

Why are each of the following statements confusing, and how would you make them clear using metric (SI) units?

a. I rode my bicycle for a distance of 15 today.

b. My dog weighs 15 .

c. It is hot today. It is 30 .

d. I lost 1.5 last week.

For the given $f(x)$, find the solution to the heat flow problem.

$$\begin{aligned} \frac{\partial u}{\partial t} &=5 \frac{\partial^2 u}{\partial x^2}, \quad 0<x<\pi, \quad t>0, \\ u(0, t) &=u(\pi, t)=0, \quad t>0, \\ u(x, 0) &=f(x), \quad 0<x<\pi, \end{aligned}$$

$f(x)=1-\cos 2 x$

Using $\iint_R \nabla^2 w d x d y=\oint_C \frac{\partial w}{\partial n} d s$, evaluate $\oint_C \frac{\partial w}{\partial n} d s$ counterclockwise over the boundary curve $C$ of the region $R$. (Show the details of your work.) $w=x^5 y+x y^5, \quad R: x^2+y^2 \leqq 1, x \geqq 0, y \geqq 0$.

Let

$$\mathbf{F}=\frac{y}{x^2+y^2} \mathbf{i}-\frac{x}{x^2+y^2} \mathbf{j}=M \mathbf{i}+N \mathbf{j}$$

(a) Show that $\partial N / \partial x=\partial M / \partial y$.

(b) Show, by using the parametrization $x=\cos t, y=\sin t$, that $\oint_C M d x+N d y=-2 \pi$, where C is the unit circle.

(c) Why doesn't this contradict Green's Theorem?

Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. A payload is dropped at an elevation of $400 \mathrm{~m}$ from a hot-air balloon that is descending at a rate of $10 \mathrm{~m} / \mathrm{s}$. Find the time when the object reaches its highest point. What is the height?

Many early balloonists dreamed of completing a trip around the world in a hot-air balloon, a goal not achieved until 1999. Write about what you imagine a trip in a balloon would be like, including a description of how manipulating air temperature would allow you to control your altitude

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?