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Sagot :
Sure, let's go through each part of the question step by step.
### Given:
- Initial cost of the laptop: [tex]\( Rs \, 75,000 \)[/tex]
- Depreciation rate per annum: [tex]\( 10\% \)[/tex]
### (i) What does [tex]\( R \)[/tex] represent in the price after [tex]\( T \)[/tex] years [tex]\( V_T = V \left( 1 - \frac{R}{100} \right)^T \)[/tex]?
In the formula [tex]\( V_T = V \left( 1 - \frac{R}{100} \right)^T \)[/tex]:
- [tex]\( V \)[/tex] is the initial cost of the laptop, which is [tex]\( Rs \, 75,000 \)[/tex].
- [tex]\( R \)[/tex] is the annual depreciation rate in percentage. In this case, [tex]\( R = 10 \% \)[/tex].
- [tex]\( T \)[/tex] is the number of years after which we want to find the depreciated value.
Therefore, [tex]\( R \)[/tex] represents the annual depreciation rate in percentage.
### (ii) What will be the price of the laptop after 2 years?
Using the formula:
[tex]\[ V_T = V \left( 1 - \frac{R}{100} \right)^T \][/tex]
For [tex]\( T = 2 \)[/tex] years:
- [tex]\( V = 75000 \)[/tex]
- [tex]\( R = 10 \)[/tex]
- [tex]\( T = 2 \)[/tex]
Substituting the values:
[tex]\[ V_2 = 75000 \left( 1 - \frac{10}{100} \right)^2 \][/tex]
[tex]\[ V_2 = 75000 \left( 0.90 \right)^2 \][/tex]
[tex]\[ V_2 = 75000 \times 0.81 \][/tex]
[tex]\[ V_2 = 60750.00000000001 \][/tex]
So, the price of the laptop after 2 years will be [tex]\( Rs \, 60750.00 \)[/tex].
### (iii) If he sold the laptop after 3 years at the same rate of compound depreciation, how much less amount would he get than if it was sold after 2 years?
First, we need to calculate the price of the laptop after 3 years:
For [tex]\( T = 3 \)[/tex] years:
- [tex]\( V = 75000 \)[/tex]
- [tex]\( R = 10 \)[/tex]
- [tex]\( T = 3 \)[/tex]
Substituting the values:
[tex]\[ V_3 = 75000 \left( 1 - \frac{10}{100} \right)^3 \][/tex]
[tex]\[ V_3 = 75000 \left( 0.90 \right)^3 \][/tex]
[tex]\[ V_3 = 75000 \times 0.729 \][/tex]
[tex]\[ V_3 = 54675.00000000001 \][/tex]
So the price of the laptop after 3 years will be [tex]\( Rs \, 54675.00 \)[/tex].
Now, let's find the difference between the price after 2 years and the price after 3 years:
[tex]\[ \text{Difference} = 60750.00000000001 - 54675.00000000001 \][/tex]
[tex]\[ \text{Difference} = 6075.00 \][/tex]
So, if he sells the laptop after 3 years, he would get [tex]\( Rs \, 6075.00 \)[/tex] less than if it was sold after 2 years.
### Given:
- Initial cost of the laptop: [tex]\( Rs \, 75,000 \)[/tex]
- Depreciation rate per annum: [tex]\( 10\% \)[/tex]
### (i) What does [tex]\( R \)[/tex] represent in the price after [tex]\( T \)[/tex] years [tex]\( V_T = V \left( 1 - \frac{R}{100} \right)^T \)[/tex]?
In the formula [tex]\( V_T = V \left( 1 - \frac{R}{100} \right)^T \)[/tex]:
- [tex]\( V \)[/tex] is the initial cost of the laptop, which is [tex]\( Rs \, 75,000 \)[/tex].
- [tex]\( R \)[/tex] is the annual depreciation rate in percentage. In this case, [tex]\( R = 10 \% \)[/tex].
- [tex]\( T \)[/tex] is the number of years after which we want to find the depreciated value.
Therefore, [tex]\( R \)[/tex] represents the annual depreciation rate in percentage.
### (ii) What will be the price of the laptop after 2 years?
Using the formula:
[tex]\[ V_T = V \left( 1 - \frac{R}{100} \right)^T \][/tex]
For [tex]\( T = 2 \)[/tex] years:
- [tex]\( V = 75000 \)[/tex]
- [tex]\( R = 10 \)[/tex]
- [tex]\( T = 2 \)[/tex]
Substituting the values:
[tex]\[ V_2 = 75000 \left( 1 - \frac{10}{100} \right)^2 \][/tex]
[tex]\[ V_2 = 75000 \left( 0.90 \right)^2 \][/tex]
[tex]\[ V_2 = 75000 \times 0.81 \][/tex]
[tex]\[ V_2 = 60750.00000000001 \][/tex]
So, the price of the laptop after 2 years will be [tex]\( Rs \, 60750.00 \)[/tex].
### (iii) If he sold the laptop after 3 years at the same rate of compound depreciation, how much less amount would he get than if it was sold after 2 years?
First, we need to calculate the price of the laptop after 3 years:
For [tex]\( T = 3 \)[/tex] years:
- [tex]\( V = 75000 \)[/tex]
- [tex]\( R = 10 \)[/tex]
- [tex]\( T = 3 \)[/tex]
Substituting the values:
[tex]\[ V_3 = 75000 \left( 1 - \frac{10}{100} \right)^3 \][/tex]
[tex]\[ V_3 = 75000 \left( 0.90 \right)^3 \][/tex]
[tex]\[ V_3 = 75000 \times 0.729 \][/tex]
[tex]\[ V_3 = 54675.00000000001 \][/tex]
So the price of the laptop after 3 years will be [tex]\( Rs \, 54675.00 \)[/tex].
Now, let's find the difference between the price after 2 years and the price after 3 years:
[tex]\[ \text{Difference} = 60750.00000000001 - 54675.00000000001 \][/tex]
[tex]\[ \text{Difference} = 6075.00 \][/tex]
So, if he sells the laptop after 3 years, he would get [tex]\( Rs \, 6075.00 \)[/tex] less than if it was sold after 2 years.
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