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 $14,000 \mathrm{gal} / \mathrm{min}$.

Given the boundary-value problem

$$\begin{array}{ll}{\frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}},} & {0<x<1, \quad 0<t<1} \\ {u(0, t)=0,} & {u(1, t)=0, \quad 0 \leq t \leq 1}\end{array}$$

$u(x, 0)=x(1-x),\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=0, \quad 0 \leq x \leq 1$ use $h=k=\frac{1}{5}$ in equation (6) to compute the values of $u_{i, 1}$ by hand.

Combine the four sentences into an effective topic sentence, or write a topic sentence accompanied by one or more supporting details.
a. Ernie sells hot dogs. b. He is weird, yet lovable. c. Everyone in town knows and likes him. d. He is part of what makes our town interesting.

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.

A clean-earth trapezoidal channel with a bottom width of 1.8 m and a side surface slope of 1:1 is to drain water uniformly at a rate of $8 m^3/s$ to a distance of 1 km, If the flow depth is not to exceed 1.2 m, determine the required elevation drop.

Find a formal solution to the given initialboundary value problem.

$$\begin{aligned} &\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}, \quad 0<x<\pi, \quad t>0 \\ &u(0, t)=0, \quad u(\pi, t)=3 \pi, \quad t>0 \\ &u(x, 0)=0, \quad 0<x<\pi \end{aligned}$$

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 hot-air balloon rises vertically from the ground so that its height after t sec is $h=\frac{1}{2} t^{2}+\frac{1}{2} t \mathrm{ft}(0 \leq t \leq 60)$

What is the velocity of the balloon at the end of 40 sec?

An analog switch uses an $n$-channel MOSFET with $V_{\mathrm{GS}(\mathrm{th})}=4 \mathrm{~V}$. A voltage of $+8 \mathrm{~V}$ is applied to the gate. Determine the maximum peak-to-peak input signal that can be applied if the drainto-source voltage drop is neglected.

In view of Clapeyron's equation and figure, is there something special about ice $I$ versus the other forms of ice ?

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.

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.

The $V_{G S(o n)}$ of an $n$-channel E-MOSFET is
a. Less than the threshold voltage
b. Equal to the gate-source cutoff voltage
c. Greater than $V_{D S(0 \pi)}$
d. Greater than $V_{G S(t h)}$