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

Question

v (t) = 700(0.76), where v is the value of the cell phone t years since its release dato.
interpret the graph of the equation and identify the true statement.
(1 point)
O As t increases, v increases slowly at first and then quickly.
O As t'increases, v decreases slowly at first and then quickly.
• As t increases, v decreases quickly at first and then slowly.
O As t increases, v increases quickly at first and then slowly.

Answer 1

The equation given for the value of the cell phone is \( v(t) = 700(0.76)^t \). This represents an exponential decay, as the value of the phone is decreasing over time at a rate of 24% per year.

To interpret the graph of the equation:

1. Initial Value: At \( t = 0 \), the value \( v(0) = 700(0.76)^0 = 700 \). This is the initial value of the cell phone when it is first released.

2. Exponential Decay: The factor \( 0.76 \) indicates that each year the value of the cell phone is multiplied by \( 0.76 \), meaning it retains 76% of its value from the previous year. This leads to a decrease in value.

3. Rate of Change: In exponential decay, the value decreases quickly at first, and as time goes on, the value approaches zero but never actually reaches it.

Thus, as \( t \) increases:

• The value \( v(t) \) decreases.
• Initially, the value decreases more rapidly, and as time passes, the decrement becomes less pronounced, meaning it decreases slowly thereafter.

Based on this, the true statement is:

• As \( t \) increases, \( v \) decreases quickly at first and then slowly.