The yearly depreciation rate for a certain vehicle is...

The yearly depreciation rate for a certain vehicle is modeled by $r=1-\left(\frac{V}{C}\right)^{1 / n}$, where $V$ is the value of the car after $n$ years, and $C$ is the original cost. a. Determine the depreciation rate for a car that originally cost $\$ 18,000$ and is worth $\$ 12,000$ after 3 yr. Round to the nearest tenth of a percent. b. Determine the original cost of a truck that has a yearly depreciation rate of $15 \%$ and is worth $\$ 11,000$ after 5 yr. Round to the nearest $\$ 100$.

The yearly depreciation rate for a certain vehicle is modeled by $r=1-\left(\frac{V}{C}\right)^{1 / n}$, where $V$ is the value of the car after $n$ years, and $C$ is the original cost.
a. Determine the depreciation rate for a car that originally cost $\$ 18,000$ and is worth $\$ 12,000$ after 3 yr. Round to the nearest tenth of a percent.
b. Determine the original cost of a truck that has a yearly depreciation rate of $15 \%$ and is worth $\$ 11,000$ after 5 yr. Round to the nearest $\$ 100$.

Key Concepts

Depreciation Rate

Depreciation rate refers to the percentage by which an asset loses its value over a certain period. It is a measure of the decline in the value of an asset and is often modeled mathematically to understand how the value decreases over time. This measurement is crucial in accounting, finance, and asset management as it impacts book values, tax deductions, and replacement planning.

Exponential Decay Models

Exponential decay models represent processes where the rate of change of a quantity is proportional to its current value. In the context of depreciation, the value of an asset declines by a consistent percentage each period, leading to a curve that decreases exponentially over time. This type of modeling is useful for accurately predicting future values or determining past values from current data.

Algebraic Manipulation and Equation Solving

Algebraic manipulation involves rearranging equations to isolate the variable of interest, which is critical when working with formulas that model depreciation. This skill encompasses understanding how to handle exponents, roots, and logarithms to solve for unknown variables. It is essential in financial mathematics to determine various parameters such as the depreciation rate or the original cost of an asset.

Rounding and Approximation

Rounding and approximation are techniques used to simplify numerical answers to a desired level of precision. In many real-world applications, such as financial calculations, results are rounded to a specific decimal place or unit to ensure clarity and practicality. This is especially important in reporting values like cost and percentages, where excessive precision may be unnecessary or unwieldy.

Transcript

Please give Ace some feedback