Question

suppose that the future price $p(t)$ of a certain item is given by the following exponential function. in this function, $p(t)$ is measured in dollars and $t$ is the number of years from today.
$p(t) = 20000(1.053)^t$
find the initial price of the item.
$\square$
does the function represent growth or decay?
\bigcirc growth
\bigcirc decay
by what percent does the price change each year?
$\square$%

Explanation:

Part 1: Find the initial price of the item

Step1: Recall the exponential function form

The general form of an exponential function is \( p(t) = a(1 + r)^t \), where \( a \) is the initial amount, \( r \) is the growth or decay rate, and \( t \) is time.
Given \( p(t) = 20000(1.053)^t \), we compare it to the general form.

Step2: Identify the initial price

In the function \( p(t) = 20000(1.053)^t \), when \( t = 0 \), \( p(0)=20000(1.053)^0 \). Since any non - zero number to the power of 0 is 1, \( p(0)=20000\times1 = 20000 \).

Step1: Recall the condition for growth/decay

For an exponential function \( p(t)=a(b)^t \), if \( b>1 \), the function represents growth; if \( 0 < b < 1 \), the function represents decay.

Step2: Analyze the base of the exponent

In the function \( p(t) = 20000(1.053)^t \), the base \( b = 1.053 \). Since \( 1.053>1 \), the function represents growth.

Step1: Recall the relationship between the base and the growth rate

For an exponential growth function \( p(t)=a(1 + r)^t \), the base \( 1 + r \) is related to the growth rate \( r \) (expressed as a decimal). Here, the function is \( p(t)=20000(1.053)^t \), so \( 1 + r=1.053 \).

Step2: Solve for the growth rate \( r \)

Subtract 1 from both sides of the equation \( 1 + r = 1.053 \). We get \( r=1.053 - 1=0.053 \).

Step3: Convert the decimal to a percentage

To convert a decimal to a percentage, we multiply by 100. So \( r = 0.053\times100\%=5.3\% \).

Answer:

The initial price of the item is \(\$20000\).

Part 2: Determine if the function represents growth or decay