Question

Tessa is opening Tessa's Terrific Tees, her own custom T-shirt business. She bought a T-shirt printing machine that is currently valued at $2,200. The machine loses
1

4
of its value every year.
Write an exponential equation in the form y=a(b)x that can model the value of the machine, y, x years after purchase.
Use whole numbers, decimals, or simplified fractions for the values of a and b.

Answer 1

To model the value of Tessa's T-shirt printing machine over time using an exponential decay equation, we can express it in the form \( y = a(b)^x \), where:

- \( y \) is the value of the machine after \( x \) years,
- \( a \) is the initial value of the machine,
- \( b \) is the decay factor,
- \( x \) is the number of years since the purchase.

1. **Initial Value (\( a \))**: The machine's initial value is $2,200. Thus, \( a = 2200 \).

2. **Decay Factor (\( b \))**: The machine loses \( \frac{1}{4} \) of its value every year. This means it retains \( 1 - \frac{1}{4} = \frac{3}{4} \) of its value each year. Hence, the decay factor \( b = \frac{3}{4} \).

Putting it all together, the equation modeling the value of the machine \( y \) after \( x \) years is:

\[
y = 2200 \left(\frac{3}{4}\right)^x
\]

So the final exponential equation is:

\[
y = 2200 \left(\frac{3}{4}\right)^x
\]