Question

suppose that the dollar value ( v(t) ) of a certain house that is ( t ) years old is given by the following exponential function. ( v(t) = 295,300(0.79)^t ) find the initial value of the house. $ does the function represent growth or decay? growth decay by what percent does the value of the house change each year? %

Explanation:

Part 1: Initial Value of the House

Step1: Recall the exponential function form

The general form of an exponential function is \( v(t) = a(b)^t \), where \( a \) is the initial value (when \( t = 0 \)).

Step2: Substitute \( t = 0 \) into the function

For \( v(t)=295300(0.79)^t \), when \( t = 0 \), we have \( v(0)=295300(0.79)^0 \). Since any non - zero number to the power of 0 is 1, \( (0.79)^0 = 1 \). So \( v(0)=295300\times1 = 295300 \).

Part 2: Growth or Decay

Step1: Recall the criteria for growth/decay

In the exponential function \( v(t)=a(b)^t \), if \( b>1 \), it is a growth function; if \( 0 < b<1 \), it is a decay function.

Step2: Analyze the value of \( b \)

In our function \( v(t)=295300(0.79)^t \), \( b = 0.79 \), and \( 0<0.79 < 1 \). So the function represents decay.

Part 3: Percentage Change per Year

Step1: Recall the formula for percentage change

For an exponential function \( v(t)=a(b)^t \), the percentage change per period is \( (b - 1)\times100\% \) (for decay, it will be negative, representing a decrease).

Step2: Calculate the percentage change

We know that \( b = 0.79 \). Then the percentage change is \( (0.79 - 1)\times100\%=( - 0.21)\times100\%=- 21\% \). The negative sign indicates a decrease, so the value of the house changes by 21% each year (a 21% decrease).

Answer:

s:

• Initial value of the house: \(\$295300\)
• The function represents: decay
• Percentage change each year: \(21\%\) (decrease, so the change is \(- 21\%\) but the question asks for the percent change, and since it's a decrease of 21%, we can say the value changes by 21% each year (with a decrease).