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The value of a cell phone decreases exponentially at a rate of 24% each year after its release date. If the initial value of th...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), 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.
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.
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GPT-4o mini
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.**
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.**
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