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Which equation best represents this graph? A - HSC - SSCE Mathematics Extension 1 - Question 7 - 2018 - Paper 1

Question 7

Which equation best represents this graph? A. P(t) = 1500 + 1500e^{-kt} B. P(t) = 3000 - 1500e^{-kt} C. P(t) = 3000 + 1500e^{-kt} D. P(t) = 4500 - 1500e^{-kt}

Worked Solution & Example Answer:Which equation best represents this graph? A - HSC - SSCE Mathematics Extension 1 - Question 7 - 2018 - Paper 1

Step 1

Identify the limiting behavior of the function

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Answer

To determine the correct equation, first analyze the graph's behavior as time progresses. It appears to approach a value asymptotically. Examining the provided equations, we find that the term $e^{-kt}$ , where $k$ is a positive constant, indicates that the function will approach a limiting value based on the additive constant.

Step 2

Match the vertical asymptote with the equations

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Answer

As time $t$ approaches infinity, the term $e^{-kt}$ tends towards zero. Therefore, we can deduce that the function will approach the additive constant of each equation. In observing the graph, the final value approached seems to be 3000, suggesting that the correct equation must have this value as its limit.

Step 3

Determine the correct option from the choices

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Answer

Based on the limiting analysis, the equation that approaches 3000 is option C: $P(t) = 3000 + 1500e^{-kt}$ , as it correctly reflects the behavior toward a limit of 3000 while maintaining the growth governed by the other term.

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