Question

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

Answer 1

The function \( v(t) = 700(0.76)^t \) represents the value of the cell phone after \( t \) years, where the initial value is $700, and the value decreases exponentially due to the 24% depreciation rate per year.

To interpret the graph of this equation, consider the behavior of the exponential decay:

1. At \( t = 0 \) (initially), the value of the phone is $700.
2. As \( t \) increases, the multiplying factor \( (0.76)^t \) decreases, which means that \( v(t) \) becomes smaller over time.
3. Since this is an exponential decay model, the value decreases quickly at first (in the initial years) because a larger percentage (24%) of the value is lost.
4. As time progresses, while the absolute dollar amount that the phone loses each year may remain significant, the percentage loss (24% of a smaller value) causes the rate of decrease in value to slow down.

Therefore, the correct interpretation is:

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

This option best describes the behavior of the cell phone’s value over time.