This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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3.2.1 To calculate the replacement value of the new machine, we use the compound interest formula for appreciation. Step 1: Identify the given values. The current cost of the machine (PV) is R278 000. The appreciation rate (i) is 7.5% per annum, or 0.075. The time period (n) is 10 years. Step 2: Apply the compound interest formula. FV = PV(1+i)^n Substitute the values: FV = 278000(1+0.075)^10 FV = 278000(1.075)^10 FV = 278000 × 2.061031569 FV = 573066.75 The replacement value of the new machine will be R573 066.75. 3.2.2 To determine the quarterly payment for the sinking fund, we first calculate the amount needed after considering the scrap value, then use the future value of an annuity formula. Step 1: Calculate the net amount needed for the new machine. The replacement value (from 3.2.1) is R573 066.75. The scrap value of the old machine is R25 000. Amount needed in sinking fund (FV_sinking) = Replacement value - Scrap value FV_sinking = 573066.75 - 25000 FV_sinking = 548066.75 Step 2: Identify the sinking fund parameters. The future value needed (FV_sinking) is R548 066.75. The interest rate (i) is 3.8% per quarter, or 0.038. The number of quarters (n) is 10 years × 4 quarters/year = 40 quarters. Step 3: Apply the future value of an annuity formula to find the quarterly payment (P). FV_sinking = P ((1+i)^n - 1)/(i) Substitute the values: 548066.75 = P (1+0.038)^40 - 10.038 548066.75 = P (1.038)^40 - 10.038 548066.75 = P (4.4530089 - 1)/(0.038) 548066.75 = P (3.4530089)/(0.038) 548066.75 = P × 90.868655 P = (548066.75)/(90.868655) P = 6031.47 The quarterly payment will be R6 031.47. 3.3 To find the equivalent annual compound interest rate, we first calculate the future value using simple interest, then use that value to solve for the compound interest rate. Step 1: Calculate the future value with simple interest. Assume a principal (P) of R1 for simplicity. The simple interest rate (i_simple) is 22% per annum, or 0.22. The time period (n) is 3 years. FV_simple = P(1 + i_simple · n) FV_simple = 1(1 + 0.22 × 3) FV_simple = 1(1 + 0.66) FV_simple = 1.66 Step 2: Set up the compound interest equation with the calculated future value. Let i_compound be the annual compound interest rate. FV_compound = P(1 + i_compound)^n 1.66 = 1(1 + i_compound)^3 Step 3: Solve for i_compound. (1 + i_compound)^3 = 1.66 1 + i_compound = (1.66)^1/3 1 + i_compound ≈ 1.184396 i_compound = 1.184396 - 1 i_compound = 0.184396 Convert to a percentage: i_compound = 0.184396 × 100\% i_compound = 18.44\% The annual rate at compound interest will be 18.44%. Drop the next question 📸