Question

Question

Use the image to answer the question.

A decreasing curve with an arrow on the end is plotted on a coordinate plane. The curve begins at the point left parenthesis 0 comma 700 right parenthesis.

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 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.
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.
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Answer 1

The equation \( v(t) = 700(0.76)^t \) describes the exponential decay of the value of a cell phone over time, where \( v \) is the value after \( t \) years. The base of the exponential function, \( 0.76 \), indicates that the value decreases by 24% each year, meaning that 76% of its previous value remains.

Based on the characteristics of exponential decay:

1. When \( t = 0 \), \( v(0) = 700(0.76)^0 = 700 \).
2. As \( t \) increases, the value \( v(t) \) decreases.
3. Initially, the value of the phone decreases quickly because a significant portion of the value is lost in the first few years.
4. Over time, as the remaining value gets smaller, the amount lost each year becomes smaller, causing the rate of decrease to slow down.

Given this understanding, the true statement is:

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