A river valley is in the shape of a V. A dam is built in this valley, so it also has a V shape, of height h and width w at the top and essentially zero at the bottom. Write an expression for the magnitude of the force that the water exerts against this dam?

Each conditional statement below is true. Write its converse. If the converse is also true, combine the statements as a biconditional. If a number is divisible by 20, then it is even.

Alex purchases x dollars worth of stock on his broker’s advice and pays his broker a 1% broker fee. The value of the shares falls to y dollars years later, and Alex uses a broker who charges 1.25% commission to make the sale. Express his net proceeds algebraically.

The bottom of a river makes a V-shape that can be modeled with the absolute value function, d(h) = 1/5|h - 240| - 48, where d is the depth of the river bottom (in feet) and h is the horizontal distance to the left-hand shore (in feet).

A ship risks running aground if the bottom of its keel (its lowest point under the water) reaches down to the river bottom. Suppose you are the harbormaster and you want to place buoys where the river bottom is 30 feet below the surface. How far from the left-hand shore should you place the buoys?

True or False In a binomial probability model, with n trials, where the probability of success is p, the expected number E of successes is .$E=np.$

Find all second partial derivatives. $h ( x , y , z ) = \frac { x ^ { 3 } e ^ { 2 y } } { z }$

At the center of the Sun, the density of the gas is

$$1.5×10^5 kg/m^3$$

and the composition is essentially 35% hydrogen by mass and 65% helium by mass. (a) What is the number density of protons there? (b) What is the ratio of that proton density to the density of particles in an ideal gas at standard temperature (0°C) and pressure (

$$1.01 × 10^5Pa$$

)?

Tell whether each function is linear, quadratic, or neither.

$$y = \frac { 3 } { 2 } x$$

Simplify each expression. If the expression does not represent a real number, say so.

$$\sqrt { \frac { 81 } { 16 } }$$

If $\tan \theta = \frac { 1 } { 2 } \text { and } \pi \leq \theta \leq \frac { 3 \pi } { 2 }$ what is the value of $\sin \theta ?$

It can be shown that the two solutions for simple harmonic oscillations,

$$x = x _ { 0 } \cos ( \omega t )$$

or

$$x = x _ { 0 } \sin ( \omega t )$$

are special cases of the more general

$$x = x _ { 0 } \cos ( \omega t + \varphi )$$

where the angle

$$\varphi$$

is known as the phase of the motion. a. State the phase of the motion when

$$x = x _ { 0 } \sin ( \omega t )$$

. b. Show explicitly that, if

$$x = x _ { 0 } \cos ( \omega t + \varphi )$$

, the period of the motion is given by

$$T = \frac { 2 \pi } { \omega }$$

independently of

$$x _ { 0 } \text { and } \varphi$$

.

Greg sets his watch for the correct time on Wednesday. Exactly one week later, he finds that his watch has lost 3¼ minutes. If his watch continues to lose time at the same rate, what will be the overall change in time after 8 weeks?

Write the verbal sentence as an inequality. Then solve the inequality and graph your solution. The product of -15 and y is less than or equal to 90.

Explain what is wrong with the statement. If P(x) and Q(x) are polynomials, P(x)/Q(x) must be continuous for all x.