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.