Problem 41 - Algebra 2 and Trigonometry

Question

A manufacturing company purchases a machine for \(\$ 50,000\) . Each year the company estimates the depreciation to be 15\(\% .\) What will be the estimated value of the machine after each of the first six years?

Step-by-Step Solution

Answer

After six years, the estimated value of the machine is \(18,857.47\).

Step 1: Understanding Depreciation

The depreciation of the machine means that its value decreases by a certain percentage each year. In this case, the machine loses 15% of its value each year due to depreciation.

Step 2: Yearly Depreciation Calculation

To find the depreciation amount, calculate 15% of the machine's current value. For the first year, the depreciation is 15% of \(50,000\). Mathematically this is calculated as \(0.15 \times 50,000 = 7,500\).

Step 3: Value After First Year

Subtract the depreciation amount from the machine's initial value to find the value after the first year: \(50,000 - 7,500 = 42,500\).

Step 4: Repeat Depreciation Calculation for Year 2

For the second year, calculate the new depreciation using the value from the end of year one. Calculate 15% of \(42,500\), which is \(0.15 \times 42,500 = 6,375\). Subtract this from \(42,500\) to find the year two value: \(42,500 - 6,375 = 36,125\).

Step 5: Continue for Year 3

Use the end-year value from year two. Calculate 15% of \(36,125\), which is \(0.15 \times 36,125 = 5,418.75\). Subtract this from \(36,125\) to find the year three value: \(36,125 - 5,418.75 = 30,706.25\).

Step 6: Year 4 Calculation

Calculate 15% of \(30,706.25\), which gives \(0.15 \times 30,706.25 = 4,605.94\). Subtract from \(30,706.25\) to find the year four value: \(30,706.25 - 4,605.94 = 26,100.31\).

Step 7: Year 5 Calculation

Calculate 15% of \(26,100.31\), giving \(0.15 \times 26,100.31 = 3,915.05\). Subtraction from \(26,100.31\) results in the year five value: \(26,100.31 - 3,915.05 = 22,185.26\).

Step 8: Year 6 Calculation

Calculate 15% of \(22,185.26\), resulting in \(0.15 \times 22,185.26 = 3,327.79\). Subtract from \(22,185.26\) to find the year six value: \(22,185.26 - 3,327.79 = 18,857.47\).

Key Concepts

Percentage Calculation

In the context of depreciation, percentage calculation plays a crucial role in determining how much value a machine loses over time. When a percentage is applied, it signifies a fraction of the whole. Imagine that we are talking about a machine that originally costs \(50,000. If it depreciates by 15% per year, we need to find out what 15% of that cost is.

To calculate 15% of \)50,000, we convert the percentage into a decimal by dividing by 100. So, 15% becomes 0.15. We then multiply this decimal by the machine's current value:

• Formula: \[ Initial ext{ }consumption ext{ }= ext{ }0.15 imes 50,000 = 7,500 \]

This calculation tells us that the first year's depreciation is $7,500. It's important to grasp this step because it gets repeated each year with a new base value as the machine's worth reduces.

Exponential Decay

Exponential decay is a process where a quantity decreases at a rate proportional to its current value. With depreciation, the concept of exponential decay becomes apparent because each year, the machine's value decreases by 15% of the remaining value rather than the original amount. Let's break this down further.

Every year, the machine's value decreases at a rate of 15%. The annual depreciation amount is subtracted from the previous year's value, decreasing the total value exponentially. Calculate it over multiple years and see this exponential pattern.

• Example: Year 1 Value Reduction: \[ 50,000 - 0.15 imes 50,000 = 42,500 \]
• Year 2 Value Reduction:\[ 42,500 - 0.15 imes 42,500 = 36,125 \]

As each year goes on, observe how the difference becomes smaller, tracing an exponential curve downwards.

Mathematical Modeling

Mathematical modeling involves creating a representation of a real-world scenario using mathematical concepts to predict future outcomes. Here, we use this to model the machine's value; it's a practical way to see how depreciation affects its worth over time.

Let's set up the model for our machine scenario:- **Initial Value**: Start with an initial amount, which in this case, is \(50,000.- **Depreciation Rate**: Use the depreciation rate of 15%, expressed as a decimal (0.15).- **Formula for Depreciated Value**: The mathematical model to find the machine's value at year "n" uses the following exponential formula:\[V = V_0 imes (1 - r)^n\]Here,

• \(V_0\): Initial value (\)50,000)
  \(r\): Depreciation rate as a decimal (0.15)
  \(n\): Number of years (1, 2, 3... 6)
• Example for Year 1:\[V_1 = 50,000 imes (1 - 0.15)^1 = 42,500\]

By plugging into this formula, anyone can model the decrease over several years without repeating detailed yearly calculations manually.