The value of a cell phone decreases exponentially at a rate of 24% each year after its release date. If the initial value of the phone is $700, the situation can be modeled by the equation

Question

v (t) = 700(0.76)*, where v is the value of the cell phone t years since its release date.
Interpret the graph of the equation and identify the true statement.
(1 point)
• As t increases, v decreases slowly at first and then quickly.
• As t increases, v increases quickly at first and then slowly.
• As t increases, v increases slowly at first and then quickly.
• As t increases, v decreases quickly at first and then slowly.

Answer 1

The equation given is

\[ v(t) = 700(0.76)^t \]

This represents an exponential decay model where the value of the cell phone decreases by 24% each year. The factor \(0.76\) indicates that each year, the cell phone retains 76% of its value from the previous year.

Interpretation of the Graph:

• At \(t = 0\), the value \(v(0) = 700(0.76)^0 = 700\). This is the initial value.
• As \(t\) increases (as years pass), the value \(v(t)\) will decrease because the base \(0.76\) is less than 1.

In an exponential decay model:

1. The value decreases quickly at first (in the initial years), as a higher percentage of the original value is lost.
2. Over time, as the value gets smaller, the dollar amount that represents 24% of the previous year's value becomes less, so the actual decrease in value slows down.

True Statement:

Based on this understanding, the correct interpretation of the graph and the true statement is:

• As t increases, v decreases quickly at first and then slowly.

This accurately describes the behavior of exponential decay where the rate of decrease slows down over time.