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Find Equivalent MARR for Two Ice-Cream Maker Models Using Present Worth Comparison
Engineering Economics
University Undergraduate
Question Content
Kiwidale Dairy is considering purchasing a new ice-cream maker. Two models, Smoothie and Creamy, are available and their information is given below. (Perform all calculation using 5 significant figures, and give your final answer to 1 decimal place.) (a) What is Kiwidale's MARR that makes the two alternatives equivalent? Use a present worth comparison. \n\n| | Smoothie | Creamy |\n| ---- | ---- | ---- |\n| First Cost | 16,000 | 39,000 |\n| Service Life | 12 years | 12 years |\n| Annual profit | 4,500 | 11,500 |\n| Annual operating cost | 1,500 | 3,720 |\n| Salvage value | 2,450 | 5,400 |
Correct Answer
18.1%
Detailed Solution Steps
1
Step 1: Define the present worth (PW) formula for each alternative. The PW of an asset is calculated as: $PW = -First\\ Cost + (Annual\\ Profit - Annual\\ Operating\\ Cost)\\times(P/A, i, n) + Salvage\\ Value\\times(P/F, i, n)$, where $(P/A, i, n)$ is the present worth factor for an annuity, and $(P/F, i, n)$ is the present worth factor for a single future amount.
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Step 2: Calculate the net annual cash flow for each model. For Smoothie: $4500 - 1500 = 3000$. For Creamy: $11500 - 3720 = 7780$.
3
Step 3: Set the PW of Smoothie equal to the PW of Creamy, since we need to find the MARR (i) where they are equivalent: $-16000 + 3000\\times(P/A, i, 12) + 2450\\times(P/F, i, 12) = -39000 + 7780\\times(P/A, i, 12) + 5400\\times(P/F, i, 12)$.
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Step 4: Rearrange the equation to group like terms: $(-16000 + 39000) = (7780 - 3000)\\times(P/A, i, 12) + (5400 - 2450)\\times(P/F, i, 12)$, simplifying to $23000 = 4780\\times(P/A, i, 12) + 2950\\times(P/F, i, 12)$.
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Step 5: Use trial and error or a financial calculator to solve for i. Test i=18%: $(P/A, 18\\%, 12)=4.7932$, $(P/F, 18\\%, 12)=0.1372$. Right-hand side (RHS) = $4780\\times4.7932 + 2950\\times0.1372 = 22911.496 + 404.74 = 23316.236$. Test i=19%: $(P/A, 19\\%, 12)=4.6105$, $(P/F, 19\\%, 12)=0.1240$. RHS = $4780\\times4.6105 + 2950\\times0.1240 = 22038.19 + 365.8 = 22403.99$.
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Step 6: Use linear interpolation to find the exact i. The target RHS is 23000. The difference between 18% and 19% RHS is $23316.236 - 22403.99 = 912.246$. The difference between target and 18% RHS is $23316.236 - 23000 = 316.236$. The fraction is $316.236/912.246 ≈ 0.3477$. So $i = 18\\% + (19\\%-18\\%)\\times(1 - 0.3477) ≈ 18.1\\%$.
Knowledge Points Involved
1
Present Worth (PW) Analysis
Present worth analysis converts all future cash flows (costs, revenues, salvage value) to their equivalent value at time 0 using a discount rate (MARR). It is used to compare investment alternatives by finding the discount rate where their present values are equal, or to select the alternative with the highest positive PW at a given MARR.
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Minimum Acceptable Rate of Return (MARR)
MARR is the minimum discount rate that a company requires to accept an investment project. In this problem, we find the MARR that makes two investment alternatives economically equivalent, meaning the company would be indifferent between choosing either model at that rate.
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Annuity and Single-Payment Present Worth Factors
$(P/A, i, n)$ is the factor used to convert a uniform annual cash flow to its present value, calculated as $\\frac{(1+i)^n - 1}{i(1+i)^n}$. $(P/F, i, n)$ converts a single future cash flow to present value, calculated as $\\frac{1}{(1+i)^n}$. These factors are core to time value of money calculations in engineering economics.
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Trial and Error with Linear Interpolation
When solving for an unknown interest rate in present worth equations, trial and error involves testing two known rates to bracket the target value, then using linear interpolation to estimate the exact rate. This method is used when algebraic solution is not feasible for complex cash flow patterns.
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