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

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.

Answer 1

The equation given, \( v(t) = 700(0.76)^t \), models the value of a cell phone where the value decreases over time. Here, the base of the exponent is less than 1 (0.76), which indicates exponential decay.

Because the value \( v(t) \) is multiplied by a fraction (0.76) raised to the power of \( t \), we can interpret the behavior of the graph as follows:

• When \( t = 0 \) (the release date), the value is $700.
• As \( t \) increases (as time goes on), the multiplier (0.76 raised to an increasing \( t \)) causes the value \( v(t) \) to decrease.
• Initially, the phone retains a significant portion of its value, but as time goes on, the percentage of value that it retains diminishes, leading to a quicker decrease in value at first, which slows down over time.

Therefore, the true statement about the graph of the equation is:

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