If the quantity of coffee-drink produced per hour (regular day hour) at a company and the costs of production are given as shown in the table below, do you think that Starbucks is experiencing economies of scale? Why or why not?   

If the price of a coffee-drink is $6, what is the quantity that maximize company profit?

The temperature at some instant (t = 0) of a laterally insulated solid rod of 1 unit length is given by ?(?,?) = sin ?? for 0  x  1, where x is measured from left end of the rod to the other end on the right. Also, at time t = 0, the left end of the rod is subjected to 0oC and maintained over the time, while the right end is insulated when t > 0. Note that the term x in the expression of the temperature is in radian. The temperature variation in the rod,

?(?,?), satisfies the heat equation: ?? ?? = ?( 2 ?^2? /??^ 2 )

Solve the above heat equation numerically with del x = 0.2 and del t = 0.04 using explicit formula: ?(?? ,??+1) = ??(??−1,??) + (1 − 2?)?(?? ,??) + ??(??+1,??)

Given that, c ^2 = 0.1 and ? = ? ^2 Δ? (Δ?)2 .

Estimate the numerical values for ?(?,?) at t = 0.08, giving your answers at 4 decimal points

1. Let X be a random variable with mean μ and variance σ . For a ∈ R, consider the expectation E ((X − a)2)
a) Write E((X −a)2) in terms of a,μ and σ2
b) For which value a is E ((X − a)2) minimal?
c) For the value a from part (b), what is E ((X − a)2)?

2. Suppose I have a group containing the following first- and second-year university students from various countries. The first 3 are male, and the last 4 female:

Name Home country Year

Andrew   UK 1

Sebastian   Germany 1

Wei China 1

Fiona UK 1

Lea Germany 2

Ajitha UK 1

Sarah UK 2

I choose a student uniformly at random from the group. Events A = ” the student is male” and B = ” the student is from the UK”. What is P(A|B)?

Rewrite the net ionic reaction. If there is NO reaction, simply write NR.

1) (CuSO₄ + NaOH)

2) (AlCl₃ + NaOH)

3) (Zn(C₂H₃O₂)₂ + NaOH)

4) (Pb(NO₃)₂ + NaOH)

5) (Na₃PO₄ + NaOH)

6) (AlCl₃ + Na₃PO₄)

7) (CuSO₄ + Na₃PO₄)

8) (Zn(C₂H₃O₂)₂ + Na₃PO₄)

An electron is trapped in a square well of unknown width, L. It starts in unknown energy level, n. When it falls to level n-1 it emits a photon of wavelength λ_photon = 2280 nm. When it falls from n-1 to n-2, it emits a photon of wavelength λ_photon = 3192 nm.

1) What is the energy of the n to n-1 photon in eV?

E_{n to n-1} =

2) What is the energy of the n-1 to n-2 photon in eV?

E_{n-1 to n-2} =

3) What is the initial value of n?

n_initial =

4) What is the width, L, of the well in nm?

L =

5) What is the longest wavelength of light, λ_longest, the well can absorb in nm?

λ_longest=

. As input you are given two arrays: an array of numbers ? and an array ? of queries where each query is a target number. The array ? is unsorted and may contain duplicates. Your goal is, for each query ? in the array ?, count the number of pairs in the array ? that sums up to ?; that is, the number of distinct pairs of indices [?, ?], with ? < ?, such that ?[?] + ?[?] = ?. Write a method, twoSumQuery, that given an array of numbers ? and an array ? of queries, returns an array ? with the same length as ?, such that for every index ?, ?[?] is the answer to query ?[?]. Let ? denote the length of ? and ? the length of ?. Suppose that in the actual input for this problem, we have the condition that ? > ? . For example, ? = 1000 and ? = 10,000,000. Your method must have time complexity ?(?).

The following is a few sample runs:

Input :     ? = [0, 1, 2, 3, 4],   ? = [1, 3, 6, 10]

Return:   [1, 2, 1, 0]

Explanation: One pair in ? sums up to 1, namely (0, 1). Two pairs in ? sum up to 3, namely (0, 3) and (1, 2). One pair in ? sum up to 6, namely (2, 4). No pair in ? sums up to 10.

public class TwoSumQuery {
   public static int[] twoSumQuery(int[] A, int[] Q) {
      
       //Replace this line with your return statement
       return null;
   }

}

*(1)(a) Find a formula for the intersection of a cone {(x,y,z): x^2+y^2=z^2} with a plane {(x,y,z): z=c}.

(b) Find a formula for the intersection of a cone {(x,y,z): x^2+y^2=z^2} with a plane {(x,y,z): x=a}.

(c) Find a formula for the intersection of a cone {(x,y,z): x^2+y^2=z^2} with a plane {(x,y,z): y=b}.

*(2) Find a formula for the intersection of a cone {(x,y,z): x^2+y^2=z^2} with a plane {(x,y,z): z=kx+b} assuming both b and k are positive.

(a) For what value of k is this an hyperbola and for what value of k is this an ellipse?

(b) Plot one of each.

Justify your answer.

Given f''(x)= 4x-6

and f'(-2)=5 and f(-2)=1

FIND: Find f'(x)=

and find f(2)=

Which alcohol would react the fastest wit Lucas Reagent?: 1-pentanol, 2-pentanol, 3-pentanol or 2-methyl-2-butanol