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 decreases quickly at first and then slowly.
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 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 equation \( v(t) = 700(0.76)^t \) represents the value of a cell phone over time, where \( t \) is the number of years since the phone's release. The term \( 0.76 \) indicates a decay factor, which reflects the 24% decrease in value each year (since \( 100% - 24% = 76% \), or 0.76).

As time progresses (i.e., as \( t \) increases), the value \( v(t) \) will decrease. The nature of exponential decay is such that:

• The value decreases quickly at first (during the initial years).
• As time goes on, the rate of decrease slows down. This is because each year's depreciation is a percentage of the remaining value rather than a fixed amount.

Therefore, the correct interpretation of the graph is:

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