Answer :
Certainly! Let's go through the problem step-by-step.
### Part 1: Future Cost of the New Delivery Truck in Five Years
1. Initial Cost: The delivery truck initially costs R450,000.
2. Annual Inflation Rate: The inflation rate is 6% per year.
3. Time Period: The time period is 5 years.
To find the future cost of the truck, we use the formula for compound interest:
[tex]\[ \text{Future Cost} = \text{Initial Cost} \times (1 + \text{Inflation Rate})^{\text{Time Period}} \][/tex]
Substituting the values:
[tex]\[ \text{Future Cost} = 450,000 \times (1 + 0.06)^{5} \][/tex]
Upon calculation:
[tex]\[ \text{Future Cost} \approx R602,201.51 \][/tex]
### Part 2: Depreciated Value of the Truck in Five Years
1. Depreciation Rate: The truck depreciates at 20% per annum.
2. Initial Cost: The initial cost remains R450,000.
3. Time Period: The time period is 5 years.
Depreciation is calculated using the reducing-balance method:
[tex]\[ \text{Depreciated Value} = \text{Initial Cost} \times (1 - \text{Depreciation Rate})^{\text{Time Period}} \][/tex]
Substituting the values:
[tex]\[ \text{Depreciated Value} = 450,000 \times (1 - 0.20)^{5} \][/tex]
Upon calculation:
[tex]\[ \text{Depreciated Value} \approx R147,456.00 \][/tex]
### Part 3: Difference to be Financed
Now, we need to find the difference between the future cost of the new truck and the depreciated value of the old truck:
[tex]\[ \text{Difference} = \text{Future Cost} - \text{Depreciated Value} \][/tex]
Substituting the values:
[tex]\[ \text{Difference} = 602,201.51 - 147,456.00 \][/tex]
Upon calculation:
[tex]\[ \text{Difference} \approx R454,745.51 \][/tex]
### Part 4: Monthly Payment for the Sinking Fund
1. Bank Interest Rate: The bank offers an interest rate of 9% per annum compounded monthly.
2. Monthly Interest Rate: The monthly interest rate is [tex]\( \frac{0.09}{12} \)[/tex].
3. Number of Months: The time period is 5 years, which is equivalent to [tex]\( 5 \times 12 = 60 \)[/tex] months.
4. Difference to be Financed: The difference, as calculated, is R454,745.51.
To find the monthly payments needed for the sinking fund, we use the sinking fund formula for compound interest:
[tex]\[ \text{Monthly Payment} = \frac{\text{Difference} \times \text{Monthly Interest Rate}}{(1 + \text{Monthly Interest Rate})^{\text{Number of Months}} - 1} \][/tex]
Substituting the values:
[tex]\[ \text{Monthly Payment} = \frac{454,745.51 \times \frac{0.09}{12}}{(1 + \frac{0.09}{12})^{60} - 1} \][/tex]
Upon calculation:
[tex]\[ \text{Monthly Payment} \approx R6,029.18 \][/tex]
### Final Summary
1. Future Cost of the New Truck: R602,201.51
2. Depreciated Value of Your Truck in Five Years: R147,456.00
3. Difference to be Financed: R454,745.51
4. Monthly Payment for the Sinking Fund: R6,029.18
These results answer the questions provided in a clear and detailed manner.
### Part 1: Future Cost of the New Delivery Truck in Five Years
1. Initial Cost: The delivery truck initially costs R450,000.
2. Annual Inflation Rate: The inflation rate is 6% per year.
3. Time Period: The time period is 5 years.
To find the future cost of the truck, we use the formula for compound interest:
[tex]\[ \text{Future Cost} = \text{Initial Cost} \times (1 + \text{Inflation Rate})^{\text{Time Period}} \][/tex]
Substituting the values:
[tex]\[ \text{Future Cost} = 450,000 \times (1 + 0.06)^{5} \][/tex]
Upon calculation:
[tex]\[ \text{Future Cost} \approx R602,201.51 \][/tex]
### Part 2: Depreciated Value of the Truck in Five Years
1. Depreciation Rate: The truck depreciates at 20% per annum.
2. Initial Cost: The initial cost remains R450,000.
3. Time Period: The time period is 5 years.
Depreciation is calculated using the reducing-balance method:
[tex]\[ \text{Depreciated Value} = \text{Initial Cost} \times (1 - \text{Depreciation Rate})^{\text{Time Period}} \][/tex]
Substituting the values:
[tex]\[ \text{Depreciated Value} = 450,000 \times (1 - 0.20)^{5} \][/tex]
Upon calculation:
[tex]\[ \text{Depreciated Value} \approx R147,456.00 \][/tex]
### Part 3: Difference to be Financed
Now, we need to find the difference between the future cost of the new truck and the depreciated value of the old truck:
[tex]\[ \text{Difference} = \text{Future Cost} - \text{Depreciated Value} \][/tex]
Substituting the values:
[tex]\[ \text{Difference} = 602,201.51 - 147,456.00 \][/tex]
Upon calculation:
[tex]\[ \text{Difference} \approx R454,745.51 \][/tex]
### Part 4: Monthly Payment for the Sinking Fund
1. Bank Interest Rate: The bank offers an interest rate of 9% per annum compounded monthly.
2. Monthly Interest Rate: The monthly interest rate is [tex]\( \frac{0.09}{12} \)[/tex].
3. Number of Months: The time period is 5 years, which is equivalent to [tex]\( 5 \times 12 = 60 \)[/tex] months.
4. Difference to be Financed: The difference, as calculated, is R454,745.51.
To find the monthly payments needed for the sinking fund, we use the sinking fund formula for compound interest:
[tex]\[ \text{Monthly Payment} = \frac{\text{Difference} \times \text{Monthly Interest Rate}}{(1 + \text{Monthly Interest Rate})^{\text{Number of Months}} - 1} \][/tex]
Substituting the values:
[tex]\[ \text{Monthly Payment} = \frac{454,745.51 \times \frac{0.09}{12}}{(1 + \frac{0.09}{12})^{60} - 1} \][/tex]
Upon calculation:
[tex]\[ \text{Monthly Payment} \approx R6,029.18 \][/tex]
### Final Summary
1. Future Cost of the New Truck: R602,201.51
2. Depreciated Value of Your Truck in Five Years: R147,456.00
3. Difference to be Financed: R454,745.51
4. Monthly Payment for the Sinking Fund: R6,029.18
These results answer the questions provided in a clear and detailed manner.
