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Postopening Scenario: Your angel investors are silent in relation to the business; however, they require board meetings. for status updates on the company's nancial health. Therefore, you need to analyze your company's performance over the last month using the data provided below. For your variance analysis, use the following financial data: Grooming Labor Standard Hours. 3150 Standard Rate $12 Actual Hours 5130 Actual Rate SHED Grooming Materials Standard Quantity 1000 Standard Price - 52 Actual Quantity 1200 Actual Price SS Prompt: Complete your work by assessing your company's nancial performance, specically addressing the following critical elements: Financial Statements o Create a statement of cost of services in the 'CDS Schedule\" tab. o Create an income statement in the \"Income Statement\" tab. "v'ariance Analysis o Identify all variances for the direct labor time and the materials price in the \"Variances\" tab. o Evaluate the significance of the variances in the 'variances' tab. MILESTONE 3 - Statement of Cost of Services INSTRUCTIONS: The following are the actual numbers for January: Materials Purchased $5,000 of Materials Consumed 40% of those purchased materials Direct Labor Direct Labor was $6,240 Overhead Overhead was 52,800 XYZ Corporation Statement of Cost of Services For the Month Ended January 31, XXXX Beginning Work in Process Inventory 0 Direct Materials: Materials - Beginning Add: Purchases for month of January Materials Available for Use Deduct: Ending Materials Materials Used Direct Labor Overhead Total Service Costs Deduct: Ending Work in Process Inventory Cost of Services * Cost of Goods Sold = Cost of Services There is no finished goods inventory to maintain.1. (11 marks) It can be shown that a vibrations in a uniform elastic beam of length L satisfy the fourth-order PDE at2 ax4 where u represents the displacement of the bar, x represents the distance along the bar, t represents time. Suppose that the ends of the bar are both kept stationary with zero displacement. (a) How many boundary and initial conditions are required to solve the PDE ? Write down the boundary conditions. (b) Use separation of variables to show that the X-equation can be written as X"" - B*X =0 and hence find the solutions: X(x) = A cos( Bx) + B sin(Bx) + C cosh(Bx) + D sinh(Bx) T(t) = a cos(cB't) + bsin(cB2t) for constants A, B, C, D, a, b and B. (c) Use the boundary conditions at x = 0 to eliminate C and D from the above expression for X(x). (d) Show that for non trivial solutions we require that Y cos( BL) cosh (L) = 1 in (Solving the above equation numerically for B gives you the to "natural modes of vibration" for the bar. Our old friend Steven of Strogatz can tell you that these have great significance when it comes to building bridges!)A simply supported uniform beam under the distributed force is shown in Fig.4. The time variation of the force is a step function. Neglect damping. a) Determine the partial differential equation governing the motion u (x, /) of the this distributed mass beam. (3 points) b) We attempt a solution of the formu(x, t) =(x)q(r), substitute it into the partial differential equation and simply the equation. (4 points) c) For a uniform beam, the general solution of (x) is (x) = C, sin Bx + C, cos Bx + C; sinh Bx + C, cosh Bx. Apply the boundary conditions for this simply supported beam and determine the natural frequency @, and corresponding modes , , make the maximum value of (x) equal to 1. (10 points) d) The total displacement is given by u(x, t) = )o, (x)q, (t) . Use the orthogonality relations to derive the EOMto this form M,q, (1) +K,q, (1) = P. (t) . Determine the M,,, K, and P. (t) by substituting the natural frequency and corresponding modes. (15 points) e) Solve the modal equations M,q, (1) + K,q, (1) = P. (). (4 points) f) Determine the deflection at mid-span u( 2., ) conly cousi at (only consider first 3 modes). (5 points) [Note] -b) Let a (t) " (x) w- and B+ = - w'm q(1) mo ( x) EI -c) The bending moment can be calculated as M = EI -d) sin ax dx = - COS aX [sin ax dx = x sin 2ax a 4a -e) The solution of the SDF which with EOM mu + ku = p(f) and boundary conditions w/ (0) = 0 & (0) =0is u(t) = (1;) 1-cos(@,t)]2.4. For a beam with both ends xed as shown in Fig. P14, the equations of motion for a three-element model are obtained as follows (these will be explained in Chaplet-7): % {3 a; % 312 o 54 -13L v1 0 E! o s E 2 \"43% 0 8L" 13L ~31} 9; a 0 L \"% \"E g 0 42 54 13L 312 o v; 0 E 2 o s -13L 3L'1 0 3L2 62 0 where 131 and 122 are the deections and 31 and 62 are the slopes at points 1 and 2, respectively. Their sign conVentions are shown in Fig. P2.4. The beam is an 8123 steel I beam with the following properties: A = 6.71 in.2 I = 64.2111} L =120in. E -= 30 x 10' psi p =_D.,000733 lb-see2 in.4 " J'. I? .. (D G) I 18 x I L l I. I L I Figure P24 damped-clamped beam modeled by three beam elements. Find the four natural frequencies (ml, mg, :3, and (:34) and plot the shapes of the two modes corresponding to all and m2.'1f the subroutine 'for solving for eigenvalues and eigenvectors of a real nonsymmetiic matrix is not available in the _ system library, use NROOT and EIGEN as described in Section 2.17. Assume that w = = 1 and that * = k. Suppose that half of the fixed cost is nonsunk. For the following production functions, solve the short- run profit maximization problem, derive the short-run supply function, and represent it graphically. On your graph, pick a price above shutdown price and graphically represent the firm's (short-run) profit at that price and at the profit-maximizing quantity. At what price does the firm make zero profits? (a) f(1, k) = VIK. The firm's short-run supply function is if p v2. 19 0 or V2 if p = v2. Here Pahutdown - V2 and at price v 2 the firm is indifferent between shutting down and producing and selling % (2 is simply up when p = v2).(b) f (1, k ) = 1aka. In this case we have SC(() = # + k, NSC(q) = + 4, ANSO(q) = & + 1, SAC(q) = 4 + 4, and SMO(q) = 2 (soc Figure 1). Recall that the firm shuts down if p > 0. The q that minimizes ANSO(q) is ka "T. It can be found by setting ANSC(q) = SMC(q) and solving for q. The minimum of ANSC(q) (i.e., Pshutdown) is 343 It can be found by plugging & into either 13 SMC(q) or ANSC(q). Given p > Pshutdown, there is only one q such that p = SMO(): 4 = V 3. ke . Therefore, if pe 3ka 43 SS(p) = kp if p > wa 43 0 or if p = 3k$ (see Figure 1)- Consider price p* in Figure 1. At p* and at the corresponding profit- maximizing quantity, call it q', the firm makes negative (short-run) profits. Profits at (p*, q') are p*q* - SC(q*) = q"(p* - SAC(q*)), so (short-run) profits at (p*, q' ) are equal to negative the blue area of the rectangle in Figure 1. The price at which (short-run) profits are zero is the minimum of SAC(q)
