A city's main well was recently found to be contaminated with trichloroethylene, a cancer-causing chemical, as a result of an abandoned chemical dump leaching chemicals into the water. A proposal submitted to city council members indicates that the cost, measured in millions of dollars, of removing x% of the toxic pollutant is given by $C(x)=\frac{0.5 x}{100-x}$ $C(x)=\frac{0.5 x}{100-x}$

Find the cost of removing 50%, 60%, 70%,80%, 90%, and 95% of the pollutant.

A manufacturer of tennis rackets finds that the total cost C(x) (in dollars) of manufacturing x rackets/day is given by $G(x)=400+4 x+0.0001 x^{2}$. Each racket can be sold at a price of p dollars, where p is related to x by the demand equation p=10-0.0004x. If all rackets that are manufactured can be sold, find the daily level of production that will yield a maximum profit for the manufacturer.

This algorithm computes $a^n$ recursively, where a is a real number and n is a positive integer. Input: a (a real number), n (a positive integer) Output: $a^n$.

$$\begin{matrix} \text{1.} & \text{exp2(a, n) \{ }\\ \text{2.} & \text{if (n == 1)}\\ \text{3.} & \text{return a}\\ \text{4.} & \ {m=\lfloor n / 2\rfloor}\\ \text{5.} & \text{power = exp2(a,m)}\\ \text{6.} & \text{power = power * power}\\ \text{7.} & \text{if (n is even)}\\ \text{8.} & \text{return power}\\ \text{9.} & \text{else}\\ \text{10.} & \text{return power * a}\\ \text{11.} & \text{ \} }\\ \end{matrix}$$

Let $b_n$ be the number of multiplications (lines 6 and 10) required to compute $a^n$. Prove that $b_{n}=\Theta(\lg n)$.

Let $p_1, . . . , p_n$ be a permutation of $\{1, \ldots, n\}$. An inversion in p is a pair $p_i, p_j$ where $i < j$ and $p_i > p_j$. Informally, the pair $p_i, p_j$ is an inversion if $p_i$ and $p_j$ are out of order, where order means increasing order. Exercise are concerned with inversions. Show that the average number of inversions in permutations of $\{1, \ldots, n\}$ is $n(n − 1)/4$. Assume that all permutations are equally likely.

The polynomial $x^{2}+1$ is prime with respect to the integers. It is not, however, prime with respect to the complex numbers. Show how $x^{2}+1$ can be factored using complex numbers.

Evaluate the expression. (-25)$^{3 / 2}$

Determine if the given system is Hamiltonian. If the system is Hamiltonian, find its Hamiltonian function.

$$\begin{array}{l} x^{\prime}=y \\ y^{\prime}=-x+x^{3} \end{array}$$

Multiply and simplify.

$$\sqrt{15} \cdot \sqrt{20}$$

Solve the equation by using the Square Root Property. $(4 m+1)^{2}=-80$

The probability that a small business will go bankrupt in its first year is 0.21. For 50 such small businesses, find the following probabilities-first by using the binomial probability formula, and then by using the normal approximation. (a) Exactly 8 go bankrupt (b) No more than 2 go bankrupt.

Find the range and standard deviation for each set of numbers. 17, 57, 48, 13, 26, 3, 36, 21, 9, 40

Find the mean for the following.

$$\begin{array}{|cc|} \hline \text { Interval } & \text { Frequency } \\ \hline 40-44 & 3 \\ \hline 45-49 & 6 \\ \hline 50-54 & 7 \\ \hline 55-59 & 14 \\ \hline 60-64 & 3 \\ \hline 65-69 & 2\\ \hline \end{array}$$

Compute the matrix exponential $e^{\mathbf{A} t}$ for each system $\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}$ given in the following problem. $x_{1}^{\prime}=5 x_{1}-4 x_{2}, x_{2}^{\prime}=3 x_{1}-2 x_{2}$

A city's main well was recently found to be contaminated with trichloroethylene, a cancer-causing chemical, as a result of an abandoned chemical dump leaching chemicals into the water. A proposal submitted to city council members indicates that the cost, measured in millions of dollars, of removing x% of the toxic pollutant is given by $C(x)=\frac{0.5 x}{100-x}$ $C(x)=\frac{0.5 x}{100-x}$

Evaluate $\lim _{x \rightarrow 100} \frac{0.5 x}{100-x}$ and interpret your result.