Answer :
- Assume the formula is $y = 20000(1-r)^2$ and set it equal to $20000(0.92)^2$.
- Simplify the equation to $(1-r)^2 = (0.92)^2$.
- Take the square root of both sides to get $1-r = 0.92$.
- Solve for $r$ and convert to percentage: $r = 1 - 0.92 = 0.08 = 8\%$. The rate of decay is $\boxed{8\%}$.
### Explanation
1. Understanding the Problem
The problem states that a boat depreciates in value according to the formula $y = 20000(0.92)^2$. We are asked to find the rate of decay and express it as a percentage. However, the given formula is not a typical depreciation formula, which usually involves a time variable. It seems the problem assumes the value of the boat after a certain period (let's say 2 years) is given by this formula. We'll assume the correct formula is $y = 20000(1-r)^2$, where $y$ is the value after 2 years and $r$ is the annual rate of decay.
2. Setting up the Equation
We are given that $y = 20000(0.92)^2$. We assume that this is the value after 2 years. So, we set up the equation $20000(1-r)^2 = 20000(0.92)^2$.
3. Simplifying the Equation
Divide both sides of the equation by 20000: $(1-r)^2 = (0.92)^2$.
4. Taking the Square Root
Take the square root of both sides: $\sqrt{(1-r)^2} = \sqrt{(0.92)^2}$, which simplifies to $1-r = 0.92$.
5. Solving for the Rate of Decay
Solve for $r$: $r = 1 - 0.92 = 0.08$.
6. Converting to Percentage
Convert $r$ to a percentage: $0.08 \times 100 = 8\%$.
7. Final Answer
Therefore, the rate of decay is 8\%.
### Examples
Understanding depreciation is crucial in managing personal and business finances. For instance, when buying a car, knowing the depreciation rate helps estimate its future value, aiding in decisions about selling or trading it in. Similarly, businesses use depreciation to account for the declining value of assets like machinery, impacting their tax liabilities and investment strategies. This concept also applies to real estate, where understanding property value depreciation can inform investment and maintenance decisions.
- Simplify the equation to $(1-r)^2 = (0.92)^2$.
- Take the square root of both sides to get $1-r = 0.92$.
- Solve for $r$ and convert to percentage: $r = 1 - 0.92 = 0.08 = 8\%$. The rate of decay is $\boxed{8\%}$.
### Explanation
1. Understanding the Problem
The problem states that a boat depreciates in value according to the formula $y = 20000(0.92)^2$. We are asked to find the rate of decay and express it as a percentage. However, the given formula is not a typical depreciation formula, which usually involves a time variable. It seems the problem assumes the value of the boat after a certain period (let's say 2 years) is given by this formula. We'll assume the correct formula is $y = 20000(1-r)^2$, where $y$ is the value after 2 years and $r$ is the annual rate of decay.
2. Setting up the Equation
We are given that $y = 20000(0.92)^2$. We assume that this is the value after 2 years. So, we set up the equation $20000(1-r)^2 = 20000(0.92)^2$.
3. Simplifying the Equation
Divide both sides of the equation by 20000: $(1-r)^2 = (0.92)^2$.
4. Taking the Square Root
Take the square root of both sides: $\sqrt{(1-r)^2} = \sqrt{(0.92)^2}$, which simplifies to $1-r = 0.92$.
5. Solving for the Rate of Decay
Solve for $r$: $r = 1 - 0.92 = 0.08$.
6. Converting to Percentage
Convert $r$ to a percentage: $0.08 \times 100 = 8\%$.
7. Final Answer
Therefore, the rate of decay is 8\%.
### Examples
Understanding depreciation is crucial in managing personal and business finances. For instance, when buying a car, knowing the depreciation rate helps estimate its future value, aiding in decisions about selling or trading it in. Similarly, businesses use depreciation to account for the declining value of assets like machinery, impacting their tax liabilities and investment strategies. This concept also applies to real estate, where understanding property value depreciation can inform investment and maintenance decisions.
