Verify that the Laplacian has the form in $\nabla^{2}=\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r} \frac{\partial}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2}}{\partial \theta^{2}}+\frac{\partial^{2}}{\partial z^{2}}$ in cylindrical coordinates.

State the base of the exponent $5$ in this expression.

$$(-3)^5$$

A hose fills a hot tub at a rate of $4.5$ gallons per minute. How many minutes will it take to fill a $400$ -gallon hot tub?

Chemists found that a complex reaction occurred in three steps. The first step takes $2.5731 \times 10^{2} s$ to complete, the second step takes $3.60 \times 10^{-1} s$, and the third step takes $7.482 \times 10^{1} s$. What is the total amount of time elapsed during the reaction? A. $3.68 \times 10^{1} \mathrm{s}$ B. $7.78 \times 10^{1} s$ C. $1.37 \times 10^{1} s$ D. $3.3249 \times 10^{2} s$

Chemical weathering is most rapid in

a. hot, dry climates.

b. cold, dry climates.

c. cold, wet climates.

d. hot, wet climates.

Choose the letter of the best answer. When magma heats underground water, the result may be a a. lava flow. b. vent. c. hot spot. d. hot spring.

In 10 minutes, the top three finishers in the 2010 Nathan’s Hot Dog Eating Contest consumed a total of 136 hotdogs. The winner, Joey Chestnut, ate 9 more hot dogs than the runner-up, Tim Janus. Pat Bertoletti finished a distant third, 8 hot dogs behind Janus. How many hot dogs did each person eat?

Write electron configurations for the following transition metal ions. $\mathrm{Ir}^{+}, \mathrm{Ir}^{3+}$

Let f(x, y) = x cos(y) and

$$x = u ^ { 2 } + v ^ { 2 }$$

and y = u - v. (a) Calculate the primary derivatives

$$\frac { \partial f } { \partial x } , \frac { \partial f } { \partial y }.$$

(b) Use the Chain Rule to calculate

$$\partial f / \partial v.$$

Leave the answer in terms of both the dependent and the independent variables. (c) Determine (x, y) for (u, v) = (2, 1) and evaluate

$$\partial f / \partial v$$

at (u, v) = (2, 1).

Springs $A$ and $B$ are identical except that $A$ is stiffer than $B$; that is, $k_A \gt k_B$ . On which spring is more work expended if they are stretched $(a)$ by the same amount and $(b)$ by the same force?

Use the Rational Zero Theorem to list all possible rational zeros for each polynomial function.

$$P(x)=6 x^5+3 x-2$$

The body falling with speed $v_0$ strikes and maintains contact with the platform supported by a nest of springs. The acceleration of the body after impact is $a=g-c y$, where $c$ is a positive constant and $y$ is measured from the original platform position. If the maximum compression of the springs is observed to be $y_m$, determine the constant $c$.

Use noble-gas notation to describe the electron configurations of the elements represented by the following symbols. Pb

S denotes the length of the arc of a circle of radius r subtended by the central angle $\theta.$ Find the missing quantity. Round answers to three decimal places. r = 2 inches, $\theta=30^{\circ},$ s = ?