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.

Question

(1 point)
Responses

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

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

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 increases slowly at first and then quickly.

Answer 1

The value of the cell phone is modeled by the equation \( v(t) = 700(0.76)^t \), which represents exponential decay.

In this equation:

• The initial value of the phone when \( t = 0 \) is $700.
• The base \( 0.76 \) indicates a decrease (since \( 0.76 < 1 \)) of 24% each year.

Looking at the behavior of the function as \( t \) increases:

• At the beginning of the time period (when the phone is new), the value decreases relatively quickly.
• As time goes on, the value continues to decrease, but the amount it decreases by each year becomes less significant in absolute terms, causing the overall decline to appear slower over time.

Thus, the correct interpretation is:

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