A school consists of $6$ separate buildings, represented by the vertices in the graph. There are paths between some of the buildings as shown. The graph also shows the length in feet of each path. School administrators want to cover some of these paths with roofs so that students will be able to walk between buildings without getting wet when it rains. To minimize cost, they must select paths to be covered such that the total length to be covered is as small as possible. Use Kruskal's algorithm to determine which paths to cover. Also determine the total length of pathways to be covered.

You are cutting off a piece of pie like this, where c is the length of the straight part (called the chord length) and b is the height of the piece.

There is an approximate formula for the area: A $\approx \frac{2}{3} c h+\frac{h^3}{2 c}$ However, h is not so easy to measure, whereas the diameter d of a pie is usually well-known. Calculate the area where the diameter of the pie is 12 inches and the chord length of the segment is 10 inches. Generalize to an algorithm that yields the area for any diameter and chord length.

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: $a^n$.

$$\begin{matrix} \text{1.} & \text{exp1(a, n) \{ }\\ \text{2.} & \text{if (n == 1)}\\ \text{3.} & \text{return a}\\ \text{4.} & \ {m=\lfloor n / 2\rfloor}\\ \text{5.} & \ {\text { return } \exp 1(a, m) * \exp 1(a, n-m)}\\ \text{6.} & \text{ \} }\\ \end{matrix}$$

Let $b_n$ be the number of multiplications (line 5) required to compute $a^n$. Find a recurrence relation and initial conditions for the sequence $\left\{b_{n}\right\}$.

Hairdryers work by blowing heat that is generated by exposed, high-resistance wires. (a) If a woman using a hairdryer comes in direct contact with the $110-\mathrm{V}$ wires, find the current that travels through her if she is dry and has a resistance of $500, 000 \Omega$ . (b) If she is wet, the resistance of her body can drop as low as $100 \Omega$. Calculate the current that passes through her body at this resistance. (Death can result when a current of $0.07$ A flows through a person's body for more than $1 \mathrm{~s}$.)

Refer to the accompanying graph. Complete parts (a)-($\text{}$c) in order. (a) Use the nearest neighbor algorithm starting at each of the vertices in turn to determine an approximate solution to the problem of finding a minimum Hamilton circuit for the graph. In each case, find the total weight of the circuit. (b) Which of the circuits found in part (a) gives the best solution to the problem of finding a minimum Hamilton circuit for the graph? (c) Just by looking carefully at the graph, find a Hamilton circuit in the graph that has lower total weight than any of the circuits found in part (a).

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: $a^n$.

$$\begin{matrix} \text{1.} & \text{exp1(a, n) \{ }\\ \text{2.} & \text{if (n == 1)}\\ \text{3.} & \text{return a}\\ \text{4.} & \ {m=\lfloor n / 2\rfloor}\\ \text{5.} & \ {\text { return } \exp 1(a, m) * \exp 1(a, n-m)}\\ \text{6.} & \text{ \} }\\ \end{matrix}$$

Let $b_n$ be the number of multiplications (line 5) required to compute $a^n$. Compute $b_2, b_3,$ and $b_4$.

Suppose that a and b are positive integers with $a \geq b .$ Let $q_{i}$ and $r_{i}$ be the quotients and remainders in the steps of the Euclidean algorithm for $i = 1, 2, . . . , n,$ where $r_n$ is the last nonzero remainder. Let

$$Q_{i}=\left(\begin{array}{ll}{q_{i}} & {1} \\ {1} & {0}\end{array}\right)$$

and $Q=\prod_{i=0}^{n} Q_{i}$. Show that

$$\left(\begin{array}{l}{a} \\ {b}\end{array}\right)=Q\left(\begin{array}{c}{r_{n}} \\ {0}\end{array}\right).$$

Give a recursive algorithm for finding gcd(a, b), where a and b are nonnegative integers not both zero, based on these facts: gcd(a, b) = gcd(b, a) if a > b, gcd(0, b) = b, gcd(a, b) = 2 gcd(a/2, b/2) if a and b are even, gcd(a, b) = gcd(a/2, b) if a is even and b is odd, and gcd(a, b) = gcd(a, b − a).

Underline any titles that should be italicized in the following sentences. If a sentence is already correct, write $C$ before the item number.

Example 1. In Madrid, I saw $\underline{\text{Guernica}}$, a fascinating painting by Pablo Picasso.

14. In the 1940's, Langston Hughes wrote a column for the New York Post in which a character called Simple voiced the thoughts of young black Americans.

Write a program that converts a Roman number such as MCMLXXVIII to its deci mal number representation. Hint: First write a function that yields the numeric value of each of the letters. Then use the following algorithm:

total = 0 While the roman number string is not empty If the first character has a larger value than the second, or the string has length 1 Add value(first character) to total. Remove the character. Else Add value(second character) - value(first character) to total. Remove both characters

The following sentence contains two personal pronouns in parentheses. Underline the pronoun that correctly completes the sentence. Then, on the line provided, write DO if the pronoun is a direct object, IO if it is an indirect object, or OP if it is an object of a preposition. Example $\underline{\text{IO}}$ 1. Mr. Graves offered Tranh and (I, $\underline{\text{me}}$) endless encouragement and guidance.

______ Norah saw (they, them) at the party for Darlene.

El cuaderno de Alberto se mojó (got wet) y ahora no puedes ver todo lo que él escribió. Ayúdalo completando con la forma correcta del pronombre posesivo. Asegúrate de que esté claro a quién o quiénes se refieren. Recuerda que debes usar la forma apropiada (el de él-el de ella, etc.).

Veo las sandalias de Gerardo, pero no las de Marta. ________ están aquí mismo. ¿Dónde estarán ________?

The cost of a phone call through this phone company depends predictably on the number of minutes you talk. The graph we saw earlier, which represents this relationship, is a line. Let t represent the number of minutes you are engaged in phone calls. Let c represent the cost of the calls. Write an algebraic equation that describes the relationship between the time in minutes and the cost of the calls. Remember that there is an up-front charge of $\$ 2.50$.

An article in Electronic Components and Technology Conference (Vol. 52, 2001) describes a study comparing single versus dual spindle saw processes for copper metallized wafers. A total of 15 devices of each type were measured for the width of the backside chipouts, $\overline{x}_{\text {single}}=66.385, s_{\text {single}}=7.895 \text { and } \overline{x}_{\text {double}}=45.278, s_{\text {double}}=8.612$ (a) Do the sample data support the claim that both processes have the same chip outputs? Assume that both populations are normally distributed and have the same variance. Answer this question by finding and interpreting the P-value for the test. (b) Construct a 95% two-sided confidence interval on the mean difference in spindle saw process. Compare this interval to the results in part (a). (c) If the $beta-error$ of the test when the true difference in chip outputs is 15 should not exceed 0.1 when $\alpha = 0.05,$ what sample sizes must be used?