Table 2 shows Excalibur Internet's demand table, total revenue, and marginal revenue at each price. Excalibur Internet's marginal cost per internet package is $47.22. What is the profit maximizing quantity and price for Excalibur Internet?

Table 2 Demand Schedule and Calculated Revenue

Price	Amount Demanded	Total Revenue	Marginal Revenue
$160	0	$0	n/a
$130	125	$16,250	$130.00
$100	250	$25,000	$70.00
$75	475	$35,625	$47.22
$40	625	$25,000	-$70.83
$0	800	$0	-$142.86

Question 3 options:

a)	125 at $130

b)	250 at $100

c)	475 at $75

d)	625 at $40

e)

800 at $0

Transcribed Image Text:

An electron is incident from a medium with effective mass m₁ onto a medium with effective electron mass m2. The difference in effective mass should cre- ate some fraction reflected and transmit some fraction. Our aim is to find the reflectivity. (a) Using the standard waveforms eikir, re-iki and teik2 with k1,2 be- ing the corresponding electron wave vector values, let us now solve for R = |r|2. The first equation is given by matching wave functions at x = 0. Write down an equation in terms of r and t. Also write down k₁ and k₂ in terms of m, E, Uo, and ħ. [5 pts] (b) In presence of a varying effective mass, the second boundary condition is a bit different from just matching wavefunction derivatives. Just like you derived this for the delta- function potential on your prac- tice problems, let us derive this once again. In presence of a varying effective mass, the Schrodinger equation looks like ħ² ə 1 (x) 2 дx m(x) дх +U(x)(x) = Ev(x) (2) Integrating the equation over a small domain around x = 0, with U(x) = 0 (since the electron is effectively free on both sides, albeit with different masses), write down an equation connecting the deriva- tives. [10 pts] (c) Use the equation connecting the derivatives to get the second equation connecting r and t [5 pts] (d) Solve the two equations to extract the reflectivity R = r². Express this in terms of the electron velocities 1,2 on either side. Recall for a free electron, v= hk/m. [5 + 5 pts] Uo An electron is incident from a medium with effective mass m₁ onto a medium with effective electron mass m2. The difference in effective mass should cre- ate some fraction reflected and transmit some fraction. Our aim is to find the reflectivity. (a) Using the standard waveforms eikir, re-iki and teik2 with k1,2 be- ing the corresponding electron wave vector values, let us now solve for R = |r|2. The first equation is given by matching wave functions at x = 0. Write down an equation in terms of r and t. Also write down k₁ and k₂ in terms of m, E, Uo, and ħ. [5 pts] (b) In presence of a varying effective mass, the second boundary condition is a bit different from just matching wavefunction derivatives. Just like you derived this for the delta- function potential on your prac- tice problems, let us derive this once again. In presence of a varying effective mass, the Schrodinger equation looks like ħ² ə 1 (x) 2 дx m(x) дх +U(x)(x) = Ev(x) (2) Integrating the equation over a small domain around x = 0, with U(x) = 0 (since the electron is effectively free on both sides, albeit with different masses), write down an equation connecting the deriva- tives. [10 pts] (c) Use the equation connecting the derivatives to get the second equation connecting r and t [5 pts] (d) Solve the two equations to extract the reflectivity R = r². Express this in terms of the electron velocities 1,2 on either side. Recall for a free electron, v= hk/m. [5 + 5 pts] Uo

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