A rectangular channel that is 10 ft wide is very smooth except for a small reach that is roughened with angle irons attached to the bottom. Water flows in the channel at a rate of 200 cfs and at a depth of 1.0 ft upstream of the rough section. Assume frictionless flow except over the roughened part, where the total drag of all roughness (all of the angle irons) is assumed to be 2000 lbf. Determine the depth downstream of the roughness for the assumed conditions.

Model each application with a differential equation. All rates are assumed to be with respect to time. The voltage drop across an inductor is proportional to the rate at which the current is changing with respect to time.

A composite beam is constructed by bolting four plates to four $60 \times$ $60 \times 12-\mathrm{~mm}$ angles as shown. The bolts are equally spaced along the beam, and the beam supports a vertical load. As proved in mechanics of materials, the shearing forces exerted on the bolts at $A$ and $B$ are proportional to the first moments with respect to the centroidal $x$ axis of the red shaded areas shown, respectively, in parts $a$ and $b$ of the figure. Knowing that the force exerted on the bolt at $A$ is $280 \mathrm{~N}$, find the force exerted on the bolt at $B$.

A beam of light in air is incident on a stack of four flat transparent materials with indices of refraction 1.20, 1.40, 1.32, and 1.28. If the angle of incidence for the beam on the first of the four materials is $60.0 ^ { \circ }$, what angle does the beam make with the normal when it emerges into the air after passing through the entire stack?

The Power Rule can be proved using implicit differentiation for the case where n is a rational number, n=p/q, and y = f(x) = x^n is assumed beforehand to be a differentiable function. If y = x^p/q, then y^q = x^p. Use implicit differentiation to show that y’=P/q x^(p/q)-1

Projectile flights are to be treated as ideal unless stated otherwise. All launch angles are assumed to be measured from the horizontal. All projectiles are assumed to be launched from the origin over a horizontal surface unless stated otherwise. For some exercises, a calculator may be helpful.

What two angles of elevation will enable a projectile to reach a target $16 \mathrm{~km}$ downrange on the same level as the gun if the projectile's initial speed is $400 \mathrm{~m} / \mathrm{sec}$ ?

What resonant frequency would you expect from blowing across the top of an empty soda bottle that is 18 cm deep, if you assumed it was a closed tube?

Show that the assumed power series expansions in Eq. we saw earlier lead to the set of equation we saw earlier through equation we saw earlier .

In Exercise below, find the derivative f'(x). In each case, the function f is assumed to be defined for all real x for which the given formula for f(x) is meaningful.

$$f(x)=\left(x+\sqrt{1+x^2}\right)^n$$

Projectile flights are to be treated as ideal unless stated otherwise. All launch angles are assumed to be measured from the horizontal. All projectiles are assumed to be launched from the origin over a horizontal surface unless stated otherwise. For some exercises, a calculator may be helpful.

A spring gun at ground level fires a golf ball at an angle of $45^{\circ}$. The ball lands $10 \mathrm{~m}$ away.

(a). What was the ball's initial speed?

True or False: Substrate controls should be collected at a fire scene from an area where it can be reasonably assumed that no flammable substance was placed. $\rule{1cm}{1pt}$.

Projectile flights in the following exercises are to be treated as ideal unless stated otherwise. All launch angles are assumed to be measured from the horizontal. All projectiles are assumed to be launched from the origin over a horizontal surface unless stated otherwise.

Flight Time and Height A projectile is fired with an initial speed of $500 \mathrm{~m} / \mathrm{sec}$ at an angle of elevation of $45^{\circ}$. How high overhead will the projectile be when it is $5 \mathrm{~km}$ downrange?

It is often assumed that changing physical quantities such as the height of a falling object or the weight of a melting snowball, are continuous functions of time. Use specific examples to discuss the merits of this assumption.

An I-beam is fabricated by welding two $16-\times 2 \mathrm{in}$. flange plates to a $24-\times 1-\mathrm{in}$. web plate. The beam is loaded in the plane of symmetry parallel to the web. On a section where the resisting moment $M_r=1400\ \mathrm{kip} \cdot \mathrm{ft}$, find the maximum flexural stress.