Question please show your work, thank you 5. An organization has N employees where N is a large number. Each employee has one of three possible job classifications and changes classifications (independently) according to a Markov chain with transition probabilities (0.7 0.2 0.1 0.2 0.6 0.2 0.1 0.4 0.5 What percentage of employees are in each classification?

​​​​​​​ 6. Let \( V=C[0,2 \pi] \) with inner product \( \langle f, g\rangle=\int_{0}^{2 \pi} f(x) g(x) d x \). Also define \( \varphi_{0}(x)=\frac{1}{\sqrt{2 \pi}} \), and for positive integers \( k \) set \( \varphi_{2 k-1}(x)=\frac{1}{\sqrt{\pi}} \cos (k x) \) and \( \varphi_{2 k}(x)=\frac{1}{\sqrt{\pi}} \sin (k x) \). (a) Show that \( \left\{\varphi_{0}, \varphi_{1}, \varphi_{2}, \ldots\right\} \) is an orthonormal set in \( V \). [Hint: Use the product-to-sum identities.] (b) Let \( f(x)=x \). Find \( \|f\| \) and \( \left\langle f, \varphi_{n}\right\rangle \) for each \( n \geq 0 \). (You don't need to give details of the integral evaluations, just the resulting values.) (c) With \( f(x)=x \), assuming that \( f(x)=\sum_{k=0}^{\infty}\left\langle f, \varphi_{k}\right\rangle \varphi_{k}(x) \), derive Leibniz's formula \( \frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots \) [Hint: Set \( x=\pi / 2 \). (d) With \( f(x)=x \), assuming that \( \|f\|^{2}=\sum_{k=0}^{\infty}\left\langle f, \varphi_{k}\right\rangle^{2} \) (see problem 10 of the midterm for why this is a reasonable statement), find the exact value of \( \sum_{k=1}^{\infty} \frac{1}{k^{2}} \). Remarks: The identity \( \|f\|^{2}=\sum_{k=0}^{\infty}\left\langle f, \varphi_{k}\right\rangle^{2} \) is known as Parseval's identity. The problem of computing the value of the infinite \( \operatorname{sum} \sum_{k=1}^{\infty} \frac{1}{k^{2}} \) is known as the Basel problem. The correct value was (famously) first found by Euler, who evaluated the sum by decomposing the function \( \frac{\sin (\pi x)}{\pi x} \) as the infinite product \( \prod_{n=1}^{\infty}\left(1-\frac{x^{2}}{n^{2}}\right) \) and then comparing the power series coefficients of both sides.

A plant bought a calciner for P220,000 and used it for 10 years, the life span of the equipment. Assume a scrap value of P20,000 for straight line method; P22,000 for textbook declining balance method and P20,000 for the double declining balance method. What is the book value of the calciner after 6 years of use?  (a) Using Straight Line Method (b) Using Declining Balance Method (c) Using Double Declining Balance Method

A solution is tested with the four indicator solutions shown in the attached colour chart. The following results are obtained: Thymol blue turns yellow. Methyl orange turns yellow. Methyl red turns orange. • Bromothymol blue turns yellow. What is the estimated pH of the solution? Thymol blue red 1,2 - 2,8 yellow 8.0-9,6 blue Methyl orange red 3,0 - 4,6 yellow Methyl red red 4,2 - 6,3 yellow Bromothymol blue yellow 6,0 - 7,8 blue 2 3 4 10 11 12 pH A. 4.3 В. 8.0 C. 5.3 D. 2.8

A 181 ‑turn circular coil of radius 2.03 cm and negligible resistance is immersed in a uniform magnetic field that is perpendicular to the plane of the coil. The coil is connected to a 16.5 Ω resistor to create a closed circuit. During a time interval of 0.135 s, the magnetic field strength decreases uniformly from 0.627 T to zero. Find the energy 𝐸 in millijoules that is dissipated in the resistor during this time interval.

The following equations describe the variation of pressure p with veloci 0 p + A og V-2 = X (i) p + B g pv = Y (iii) p + CPV 2 = Z where, A, B, and C are dimensionless constants. X, Y, and Z are values with the same dimensions as pressure, g is the acceleration due to gravity, p is the density of the liquid, and o is the surface tension of the liquid. Which equations are dimensionally homogeneous? 2:33 PM

The table below shows the monthly cost of producing vintage model cars for collectors for quantities O, 100, 200, and 500. Complete the table by filling in the average fixed cost, average variable cost, and average total cost. Instructions: Round your answers to 2 decimal places. Average Fixed Cost (dollars) Average Variable Cost (dollars) Average Total Cost (dollars) Vintage Model Car Production Costs Total Fixed Total Variable Output Cost (dollars) Cost (dollars) 0 $1,500 $0 100 1,500 600 200 1,500 1,100 500 1,500 4,150 Total Cost (dollars) $1,500 2,100 2,600 5,650