Use the image to answer the question.

Question

A decreasing curve with an arrow on the end is plotted on a coordinate plane. The curve begins at the point left parenthesis 0 comma 700 right parenthesis.

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 v(t)=700(0.76)t
, 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)
Responses

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

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

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

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

Answer 1

The situation you've described involves an exponential decay of the cell phone's value. Given the equation \( v(t) = 700(0.76)^t \), we can interpret the behavior of the value \( v \) over time \( t \).

In this case, as \( t \) (the number of years since the phone's release date) increases, the value \( v(t) \) will decrease because the base of the exponent (0.76) is less than 1. This indicates that the value is decreasing over time.

The key characteristic of exponential decay is that it decreases quickly at the beginning and then more slowly as time goes on. This means that a significant portion of the value is lost in the first few years, and then as time continues, the value diminishes at a slower rate.

Given the options, the correct interpretation of the graph is:

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