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Chapter 10: Problem 43

. A property decreases in value by \(0.5 \%\) each week. By what percent does it decrease after one year (52 weeks)?

Short Answer

Expert verified
Answer: The property's value decreases by approximately 24.78% after one year (52 weeks).

Step by step solution

01

Understand the problem

We are given that the value of a property decreases by \(0.5\%\) each week. We need to find the percent decrease in value after one year which consists of 52 weeks.
02

Express the percent decrease formula

We first need to find the weekly decrease factor. The formula for calculating the final value (V) after decreasing an initial value (P) by a percentage (d) is: $$V = P - P \cdot d$$ If we represent the weekly decrease factor as x, the final value after one week is given by: $$V = P\cdot x$$ Equating these two expressions, we have: $$P - P \cdot d = P\cdot x$$ We now need to find the x value, and then use it to find the overall percent decrease after a year.
03

Find the decrease factor

Solving for x in the above equation, we get: $$x = 1 - d$$ The problem tells us that the value decreases by \(0.5\%\), so we have: $$d = 0.5\% = \frac{0.5}{100} = 0.005$$ Hence, the decrease factor is: $$x = 1 - 0.005 = 0.995$$
04

Apply the exponential decay formula

After 52 weeks (one year), we can calculate the total decrease factor after n weeks (in this case, n = 52) using the decrease factor x: $$D = x^n$$ $$D = 0.995^{52}$$ $$D \approx 0.7522$$
05

Calculate the overall percent decrease

Now, we want to find out the total percent decrease. By subtracting D from 1 and then multiplying by 100, we can obtain the total percent decrease: $$\begin{aligned} \text{Percent Decrease} &= (1 - D) \cdot 100 \\ &= (1 - 0.7522) \cdot 100 \\ &\approx 24.78 \% \end{aligned}$$ Therefore, the property's value decreases by approximately \(24.78\%\) after one year (52 weeks).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Decay
Exponential decay is a process where a quantity decreases at a constant rate over time. This is often discussed in scenarios involving assets losing value, or materials breaking down. In the context of property value, exponential decay helps describe how a property鈥檚 market worth can reduce over given intervals, such as weeks, months, or years. This is not a linear decrease, meaning it doesn't drop by the same absolute amount each period but rather reduces relative to its remaining value.\[ V = P imes x^n \]- **V:** Final value after time - **P:** Initial value- **x:** Weekly decrease factor- **n:** Number of time periods (e.g., weeks)By understanding exponential decay, you can anticipate how investments like property may perform over time, making this concept invaluable for long-term financial planning.
Weekly Decrease Factor
The weekly decrease factor in the context of property value is a multiplier representing one week鈥檚 percent decrease. This factor is important for calculating the property value over extended periods. To find the decrease factor, consider the percent decrease transformed into a decimal and subtract it from 1:\[ x = 1 - d \]- **x:** Weekly decrease factor - **d:** Weekly percent decrease (in decimal form)For instance, if the property decreases by 0.5% each week, then:\[ d = \frac{0.5}{100} = 0.005\]Thus, the weekly decrease factor is:\[x = 1 - 0.005 = 0.995\]Over time, this factor works sequentially to move the value lower, simulating a gradual decline typical in exponential decay.
Decrease Formula
The decrease formula allows you to calculate a property鈥檚 value after a certain number of weeks if the value is known to decay by a consistent percentage each week. It encapsulates the ideas of both exponential decay and the weekly decrease factor.The formula is:\[ V = P \times x^n \]- **V:** Value after time - **P:** Initial property value- **x:** Weekly decrease factor- **n:** Number of weeksBy applying this formula, you can determine the residual value of properties subjected to regular depreciation and reliably project future values.
Property Value Depreciation
Property value depreciation is the concept of a piece of real estate losing value over time. This can be due to wear and tear, market conditions, or other factors. Understanding depreciation is essential for property investors, as it influences calculations on investment returns, taxes, and resale value. The calculations of decay or depreciation are often expressed in percentages, helping investors track and anticipate loss in value over specified periods. By applying the concepts of exponential decay and the decrease formula, investors can strategize to minimize losses:
  • Estimate the future value of properties.
  • Plan for tax implications connected to lower values.
  • Make informed decisions about when to sell or renovate.
Tracking depreciation is crucial in maintaining profitable investments and avoiding undue financial setbacks.

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Most popular questions from this chapter

Investment \(A\) grows at an annual rate of \(r=10 \%\), and investment \(B\) grows at a continuous rate of \(k=\) \(10 \%\). Compare these two investments, both initially worth \(\$ 1000\), over a 50-year period by completing Table 10.12

In year \(t,\) the population, \(L,\) of a colony of large ants is \(L=2000(1.05)^{t}\), and the population of a colony of small ants is \(S=1000(1.1)^{t}\). (a) Construct a table showing each colony's population in years \(t=5,10,15,20,25,30,35,40\). (b) The small ants go to war against the large ants; they destroy the large ant colony when there are twice as many small ants as large ants. Use your table to determine in which year this happens. (c) As long as the large ant population is greater than the small ant population, the large ants harvest fruit that falls on the ground between the two colonies. In which years in your table do the large ants harvest the fruit?

For the investments described, assume that \(t\) is the elapsed number of years and that \(T\) is the elapsed number of months. (a) Describe in words how the value of the investment changes over time. (b) Give the annual growth rate. \(V=625(1.03)^{T / 3}\)

For the functions in the form \(P=a b^{t / T}\) describing population growth. (a) Give the values of the constants \(a, b\), and \(T\). What do these constants tell you about population growth? (b) Give the annual growth rate. \(P=400\left(\frac{2}{3}\right)^{t / 14}\)

A population is initially 15,000 and grows at a continuous rate of \(2 \%\) a year. Find the population after 20 years.
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