Idea Transcript
IE 343 Section 1 Engineering Economy Exam 1 Review Problems – Solution Instructor : Tian Ni Feb. 10, 2012 1. A firm is planning to manufacture a new product. The sales department estimates the unit selling price is P=$35-0.02D. The fixed cost of manufacturing and selling the product is $8,000 and the variable cost per each unit sold is $4. a. Suppose the firm wants to reach a profit of $3,900. How many units do they need to sell in order to achieve that? (Notice that there may be more than one choice for the company, and you are expected to report them all) We want Profit (D) = 3,900
Therefore, if the firm sells 700 units or 850 units, it will realize a profit of $3,900. b. What is the number of units that the firm has to sell in order to achieve the maximum profit?
c.
What is the firm’s maximum possible revenue?
Revenue= 35 D-0.02 D2
Maximum Revenue = Revenue(875) = 35 * 875 – 0.02 * 875 *875 = $15,312.5
2. Calculate the equivalent uniform annual cost of the following schedule of payments. Let: i=18% 400 100
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500
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500
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(a) Decompose the cash flow diagram into several Basic Components. 100
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14 15
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14 15
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(b) Calculate the uniform equivalent annual cost
of the cash-flow diagram.
Method 1: Based on the cash flow decomposition in part(a), we can first find out the present equivalent value P of the basic components and then find out the annual equivalent cost A P = 100(P/A,18%, 15) + 100(P/G, 18%, 5) + 100(P/G, 18%, 5)(P/F, 18%, 5) + 100(P/G, 18%, 5)(P/F, 18%, 10) = 1360.9 A = 1360.9(A/P,18%,15) = 267.28 Method 2: Since cash flows repeat every 5 years, the annual equivalent cost for the first 5 years period is the same as the annual equivalent cost for the whole 15 years period. So we can analyze for 5 years only. Therefore, we get A = 100 + 100 (A/G, 18%, 5) = 100+100*1.6728=267.28
3. What is the value of K that makes the 2 cash flows equivalent? Assume i = 12% a year.
5K 4K 2K
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250
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500
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750
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1000
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1250
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(a) Write an expression for that makes the two cash-flow diagrams equivalent to each other. should be the only variable on the left side of the equation (i.e., ). You don’t need to calculate the final numerical answer.
Present value of cash flow 1 = 5K (P/F, 0.12, 1) + 2K (P/F, 0.12, 4) + 4K (P/F, 0.12, 6) Cash flow 2 consists of a standard annuity of $250 paid for 5 years and a standard 5 years uniform gradient series with G = $250. Present value of cash flow 2 = 250 (P/A, 12%, 5) + 250 (P/G, 12%, 5) Present value of cash flow 1 = Present value of cash flow 2 5K (P/F, 0.12, 1) + 2K (P/F, 0.12, 4) + 4K (P/F, 0.12, 6) = 250 (P/A, 12%, 5) + 250 (P/G, 12%, 5) K = [250 (P/A, 12%, 5) + 250 (P/G, 12%, 5)] / [5 (P/F, 0.12, 1) + 2 (P/F, 0.12, 4) + 4 (P/F, 0.12, 6)]
(b) Calculate the numerical answer for
based on your expression in (a).
Present value of cash flow 1 = 5K/(1.12) + 2K / (1.12)4 + 4K / (1.12)6 = 7.7618K Present value of cash flow 2 = 250 (P/A, 12%, 5) + 250 (P/G, 12%, 5) = 250 (3.6048) + 250 (6.397) = $2500.45 Therefore, 7.7618K = 2500.45 K = $322.15
4. Suppose you have a bank account that pays 6% annual interest compounded monthly. Answer the following questions a. If you deposit $100 into the account at the end of every month starting from the end of month 1 for the next 4 years, how much will you have in the account at the end of the 4th year? The payment period is in month, so we have to find out the effective monthly interest rate. Since the 6% interest is compounded monthly, so the nominal interest rate per month is the same as effective interest rate per month = 6% / 12= 0.5% In 4 years, we make totally 4*12 = 48 deposits. F = 100(F/A, 0.5%, 48) = 5409.8
b. If you deposit $100 into the account at the end of every 6 months for the next 4 years, how much will you have in the account at the end of the 4th year? Write an expression only. The payment period is in 6 months, so we have to find out the effective interest rate per 6 months. The nominal interest rate per 6 months is 6% / 2 = 3% The effective interest rate per 6 months is In 4 years, we make totally 4*2 = 8 deposits. F = 100(F|A, 3.04%, 8)
c. Say you deposit $100 at the end of the first quarter, $200 at the end of the second quarter, $300 at the end of the third quarter, and so on for 4 years. Every quarter you deposit $100 more than the previous quarter. How much will you have in the account at the end of the 4th year? Write an expression only.
First draw the cash flow diagram from the banks prospective
400 300 200 100
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The payment period is in quarters, so we have to find out the effective interest rate per quarter. The nominal interest rate per quarter is 6% / 4 = 1.5% The effective interest rate per quarter is In 4 years, we make totally 4*4 = 16 deposits. To calculate the F of the cash flow diagram, we make a standard decomposition of the cash flow diagram into a standard annuity with A = 100 for 16 periods and a standard uniform gradient series of G = 100. F = 100(F/A, 1.51%, 16) + 100(F/G, 1.51%, 16)