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

Question

, 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 decreases slowly at first and then quickly.
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 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 increases slowly at first and then quickly.

Answer 1

The value of the cell phone is modeled by the equation \( v(t) = 700(0.76)^t \). In this model:

• The initial value of the phone is $700.
• Each year, the value decreases by 24%, which means it retains 76% of its value from the previous year (hence the \( 0.76 \) in the equation).

The graph of this equation represents an exponential decay.

As \( t \) increases (which signifies time passing after the phone's release), the value \( v(t) \) will decrease.

1. At first, when \( t \) is small (for the first few years), the decrease is more noticeable than later on because the value is still relatively high.
2. As time passes, the value approaches zero, but the amount of decrease each year becomes smaller in absolute terms, even though it's a constant percentage (24%).

This means:

• The cell phone’s value decreases quickly at first when it is worth more, and then the decrease begins to slow down as the value decreases toward zero.

Thus, the true statement is:

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