Four equal charges of

$$5.0 \mu \mathrm { C }$$

are placed at the vertices of a square of side 10 cm. a. Calculate the value of the electric potential at the centre of the square. b. Determine the electric field at the centre of the square. c. How do you reconcile your answers to a and b with the fact that the electric field is the derivative of the potential?

Find each limit algebraically. $\lim _{x \rightarrow-1}\left(8 x^{5}-7 x^{3}+8 x^{2}+x-4\right)$

Write and solve a proportion for each situation. A daffodil that is 1.2 feet tall casts a shadow that is 1.5 feet long. At the same time, a nearby lamppost casts a shadow that is 20 feet long. What is the height of the the lamppost?

Lane 1 at your local track is 0.25 mi long. You live 0.5 mi away from the track. Which of the following results in the shortest jog? F. jogging 6 times around the track in Lane 1 G. jogging to the track and then 5 times around the tracking Lane 1 H. jogging to the track, 3 times around the track in Lane 1, and then home I. jogging 8 times around the track in Lane 1

Say whether l’Hospital’s rule applies. If is does, use it to evaluate the given limit. If not, use some other method. $\lim _{x \rightarrow-\infty} \frac{x^{2}+30 x}{2 x^{6}+10 x}$

Process Safety Progress (Dec. 2004) published a risk analysis for a natural-gas pipeline between Bolivia and Brazil. The most likely scenario for an accident would be natural-gas leakage from a hole in the pipeline. The probability that the leak ignites immediately (causing a jet fire) is .01. If the leak does not immediately ignite, it may result in the delayed ignition of a gas cloud. Given no immediate ignition, the probability of delayed ignition (causing a flash fire) is .01. If there is no delayed ignition, the gas cloud will disperse harmlessly. Suppose a leak occurs in the natural-gas pipeline. Find the probability that either a jet fire or a flash fire will occur. Illustrate with a tree diagram.

Suppose

$$\Delta A B C \sim \Delta T U V.$$

Determine whether each pair of measures is equal. the ratios of the sides

$$\frac { B C } { U V } \text { and } \frac { A C } { T V }$$

Find the derivative of the following functions. y=$\sqrt { f ( x ) }$, where f is differentiable and nonnegative at x

Evaluate the given iterated integral. $\int_{0}^{4} \int_{0}^{1 / 2} \int_{0}^{x^{2}} \frac{1}{\sqrt{x^{2}-y^{2}}} d y d x d z$

Solve the equation. Check to justify your answer. -2a - 8 + 5a = 19

Use a graphing utility to graph the function. What do you observe about its asymptotes? $g(x)=\frac{3 x^{4}-5 x+3}{x^{4}+1}$

Describe the sampling distribution of $\hat{p}$. Assume that the size of the population is 25,000 for each problem. n=300, p=0.7

A set S of real numbers is closed under addition if the sum of any two members of S is in S. A set S is closed under multiplication if the product of any two members of S is in S. For each set given below tell whether the set is closed under addition. If the set is not closed, give an example to show this. The irrational numbers

The diameters of the pistons shown in figure are $D_{1}=3$ $\mathrm{in}$ and $D_{2}=2$ $\mathrm{in}$. Find the pressure in chamber $3$ , in psia, when the other pressures are $P_{1}=150$ $\mathrm{psia}$ and $P_{2}=200$ $\mathrm{psia}$.