Suppose that the dollar value [tex]$v(t)$[/tex] of a certain house that is [tex]$t$[/tex] years old is given by the following exponential function:
[tex]$v(t)=476,000(0.87)^t$[/tex]
Find the initial value of the house.
$476000
Does the function represent growth or decay?
growth decay
By what percent does the value of the house change each year?
$\square$ %
Answer (1)
The initial value of the house is found by evaluating the function at t = 0 , which gives $476,000.
Since the base of the exponential function is 0.87 , which is between 0 and 1, the function represents decay.
The percentage change in value each year is calculated as ( 0.87 − 1 ) × 100 = − 13% .
Therefore, the value of the house decreases by 13% each year, and the initial value is $476 , 000 .
Explanation
Understanding the Problem The problem provides an exponential function that models the value of a house over time. We need to find the initial value, determine if it represents growth or decay, and calculate the annual percentage change in value.
Finding the Initial Value To find the initial value, we need to find the value of the house when t = 0 . We substitute t = 0 into the function: v ( 0 ) = 476 , 000 ( 0.87 ) 0 Since any number raised to the power of 0 is 1, we have: v ( 0 ) = 476 , 000 × 1 = 476 , 000 So, the initial value of the house is $476,000.
Determining Growth or Decay To determine if the function represents growth or decay, we look at the base of the exponential function, which is 0.87 . Since 0 < 0.87 < 1 , the function represents exponential decay. This means the value of the house decreases over time.
Calculating Percentage Change To find the percentage change in value each year, we look at the base again, 0.87 . This represents the proportion of the value that remains each year. To find the percentage change, we subtract 1 from the base and multiply by 100: ( 0.87 − 1 ) = − 0.13 − 0.13 × 100 = − 13 So, the value of the house decreases by 13% each year.
Examples
Understanding exponential decay is crucial in finance, especially when dealing with depreciation. For instance, when a company buys equipment, its value decreases over time. If a company buys a machine for $50,000 and it depreciates at a rate of 15% per year, we can use an exponential decay model to predict its value after a certain number of years. This helps in financial planning, tax calculations, and making informed decisions about when to replace the equipment. Similarly, understanding the depreciation of a car's value can help you decide when it's best to sell or trade it in.
