The value V of a machine t years after it is purchased is inversely proportional to the square root of t+2. The initial value of the machine is $10,000. Find the rate of depreciation when t=5 . Round your answer to two decimal places.
A. -381.80 PER YEAR
B. -1889.82 PER YEAR
C. 447.21 PER YEAR
D. 1767.77 PER YEAR
E. -763.60 PER YEAR
E. -763.60 PER YEAR
To find the rate of depreciation when t=5, we need to find the derivative of V with respect to t and then substitute t=5 into the derivative.
Given that V is inversely proportional to the square root of t+2, we can express this relationship as:
V = k / √(t+2)
where k is a constant.
To find k, we can use the initial value of the machine, V(0) = $10,000. Substituting t=0 into the equation, we get:
10,000 = k / √(0+2)
10,000 = k / √2
Squaring both sides of the equation, we get:
100,000,000 = k^2 / 2
k^2 = 200,000,000
k = √200,000,000
k ≈ 14,142.14
Now we have an equation for V as a function of t:
V = 14,142.14 / √(t+2)
To find the rate of depreciation when t=5, we need to find dV/dt and substitute t=5 into the derivative:
dV/dt = [d/dt (14,142.14)] / √(t+2)
The derivative of a constant is zero, so the derivative simplifies to:
dV/dt = 0 / √(t+2)
dV/dt = 0
Therefore, the rate of depreciation when t=5 is zero. None of the given answer choices are correct.
To find the rate of depreciation when t=5, we need to first find the value of the machine when t=5 and then calculate the difference between the initial value and the value at t=5, divided by the difference in time.
Given that the value V of the machine is inversely proportional to the square root of t+2, we can express this relationship mathematically as:
V = k / √(t+2)
Where k is the constant of proportionality.
To find the value of k, we can substitute the initial value of the machine (V= $10,000) when t=0 into the equation:
$10,000 = k / √(0+2)
Simplifying the equation, we get:
$10,000 = k / √2
To isolate k, we multiply both sides of the equation by √2:
$10,000 * √2 = k
k ≈ $14,142.14
Now we can substitute the value of k into the equation for V:
V = $14,142.14 / √(t+2)
To find the value of the machine when t=5, we substitute t=5 into the equation:
V = $14,142.14 / √(5+2)
V ≈ $4,048.37
To find the rate of depreciation, we calculate the difference between the initial value ($10,000) and the value at t=5 ($4,048.37), divided by the difference in time (5 years):
Rate of depreciation = ($10,000 - $4,048.37) / 5
Rate of depreciation ≈ $1950.73 / year
Rounding the answer to two decimal places, the rate of depreciation when t=5 is approximately $1950.73 per year.
None of the given answer options match this result, so none of the options A, B, C, D, or E is correct.