A company buys an office machine for $5200 on January 1 of a given year. the machine is expected to last for eight years at the end of which time its trade in value or salvage value will be 1100$. if the company figures the decline in value to be the same each year than the book value or salvage value after T years ,0<t<8 is given by the linear function: v(t) =C-t (c-s/n)
Where c =the original cost of the item($5200), N=the number of years of expected life (8)& s=the salvage value ($1100)
A.) find the linear function for the straight line depreciation of the office machine
B.) find the salvage value after 0 years, 3years and 7 years.
My work: a.) v(t) =5200-t(5200-1100/8)
I don't think if this is correct which is why I cannot move on to part b
To find the linear function for the straight-line depreciation of the office machine, you need to use the formula v(t) = C - t((C - s) / N), where:
- C represents the original cost of the item ($5200),
- t represents the number of years since the purchase (0 < t < 8),
- s represents the salvage value ($1100),
- N represents the number of years of expected life (8).
Now let's plug in the values into the formula:
v(t) = 5200 - t((5200 - 1100) / 8)
Simplifying the expression within the parentheses:
v(t) = 5200 - t(4100 / 8)
Simplifying further:
v(t) = 5200 - 512.5t
This is the linear function for the straight-line depreciation of the office machine.
Now we can move on to part B and find the salvage value after 0 years, 3 years, and 7 years.
To find the salvage value after 0 years, we substitute t = 0 into the function:
v(0) = 5200 - 512.5(0)
v(0) = 5200
The salvage value after 0 years is $5200.
To find the salvage value after 3 years, substitute t = 3 into the function:
v(3) = 5200 - 512.5(3)
v(3) = 5200 - 1537.5
v(3) = 3662.5
The salvage value after 3 years is $3662.5.
To find the salvage value after 7 years, substitute t = 7 into the function:
v(7) = 5200 - 512.5(7)
v(7) = 5200 - 3587.5
v(7) = 1612.5
The salvage value after 7 years is $1612.5.
To find the linear function for the straight line depreciation, you need to use the given formula: v(t) = c - t(c - s/n), where c is the original cost ($5200), n is the number of years of expected life (8), t is the number of years since the purchase, and s is the salvage value ($1100).
a) Substituting the given values into the formula:
v(t) = 5200 - t(5200 - 1100/8)
Simplifying further:
v(t) = 5200 - t(5200 - 137.5)
v(t) = 5200 - (5200t - 137.5t)
v(t) = 5200 - 5200t + 137.5t
v(t) = 5200 - 5062.5t
Therefore, the linear function for the straight line depreciation of the office machine is v(t) = 5200 - 5062.5t.
b) To find the salvage value after 0 years, 3 years, and 7 years, substitute the respective values of t into the linear function:
For t = 0:
v(0) = 5200 - 5062.5(0) = 5200 - 0 = 5200
The salvage value after 0 years is $5200.
For t = 3:
v(3) = 5200 - 5062.5(3) = 5200 - 15187.5 = -9997.5
The salvage value after 3 years is -$9997.5 (which means that it has depreciated more than the original cost).
For t = 7:
v(7) = 5200 - 5062.5(7) = 5200 - 35437.5 = -30237.5
The salvage value after 7 years is -$30237.5 (which means that it has depreciated significantly).
Note: Negative salvage values indicate that the machine's value has depreciated beyond the expected salvage value.