The value V of a machine $t$ years after it is purchased is inversely proportional to the square root of $t + 5$ . The initial value of the machine is $\$$ 10,000. Find the rate of depreciation when $t = 3$ . Round your answer to two decimal places.
A)-494.11 per year
B)-1767.77 per year
C)962.25 per year
D)447.21 per year
E)-988.21 per year

Question

Quizplus · Accepted Answer

The value $V$ of the machine is inversely proportional to the square root of $t + 5$, so we can write $V = \frac{k}{\sqrt{t + 5}}$, where $k$ is a constant. When $t = 0$, $V = 10000$, so $10000 = \frac{k}{\sqrt{0 + 5}}$, which gives $k = 10000\sqrt{5}$. To find the rate of depreciation at $t = 3$, we differentiate $V$ with respect to $t$: $\frac{dV}{dt} = -\frac{k}{2(t + 5)^{3/2}}$. Substituting $k = 10000\sqrt{5}$ and $t = 3$, we get $\frac{dV}{dt} = -\frac{10000\sqrt{5}}{2(3 + 5)^{3/2}} = -494.11$, which means the machine depreciates at a rate of $-494.11$ per year at $t = 3$.

The Value V of a Machine ttt Years After It Is Purchased Is Inversely Proportional to the Is

The Value V of a Machine $t$ Years After It Is Purchased Is Inversely Proportional to the Is