Question

An annual depreciation rate is the percent that the value of an item decreases each year. A company purchases technology for $5,000. The company uses the function $r=1-\sqrt[3]{\frac{S}{5000}}$

to relate the annual depreciation rate $r$ (in decimal form) and the value $S$ (in dollars) of the technology after 3 years. Find $S$ when $r=0.15$ .

Answer 1

We are given the annual depreciation rate function $r=1-\sqrt[3]{\frac{S}{5000}}$, where $S$ is the current value of the technology. We want to find the value of $S$ after 3 years, when $r=0.15$.

First, we can solve the equation $r=1-\sqrt[3]{\frac{S}{5000}}$ for $S$, to get:

$\sqrt[3]{\frac{S}{5000}} = 1-r$

Cubing both sides, we get:

$\frac{S}{5000} = (1-r)^3$

Multiplying both sides by 5000, we get:

$S = 5000(1-r)^3$

Now, we can substitute $r=0.15$ into this equation, to get:

$S = 5000(1-0.15)^3$

Simplifying:

$S = 5000(0.85)^3$

$S = 5000(0.614125)$

$S = 3070.63$

Therefore, the value of the technology after 3 years is $S = \$3070.63$.