Questions: Exponential Growth Given that a quantity Q(t) is described by the exponential growth function Q(t)=500 e^(0.02 t) where t is measured in minutes, answer the following questions. (a) What is the growth constant k ? k= (b) What quantity is present initially? units (c) Complete the following table of values. (Round your answer to the nearest integer.) t Q ------ 0 10 20 100 1000

Exponential Growth Given that a quantity Q(t) is described by the exponential growth function Q(t)=500 e^(0.02 t) where t is measured in minutes, answer the following questions. (a) What is the growth constant k ? k= (b) What quantity is present initially? units (c) Complete the following table of values. (Round your answer to the nearest integer.) t Q ------ 0 10 20 100 1000
Transcript text: Exponential Growth Given that a quantity $Q(t)$ is described by the exponential growth function \[ Q(t)=500 e^{0.02 t} \] where $t$ is measured in minutes, answer the following questions. (a) What is the growth constant $k$ ? $k=$ $\qquad$ (b) What quantity is present initially? $\qquad$ units (c) Complete the following table of values. (Round your answer to the nearest integer.) \begin{tabular}{|c|c|} \hline $\boldsymbol{t}$ & $\boldsymbol{Q}$ \\ \hline 0 & \\ \hline 10 & \\ \hline 20 & \\ \hline 100 & \\ \hline 1000 & \\ \hline \end{tabular}
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Solution

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Solution Steps

Step 1: Identify the Initial Quantity

The initial quantity \(Q_0\) given is 500.

Step 2: Determine the Growth (or Decay) Constant

The growth (or decay) constant \(k\) given is 0.02.

Step 3: Calculate the Quantity \(Q(t)\)

Using the formula \(Q(t) = Q_0 e^{kt}\), where \(t = 0\), we calculate: \(Q(0) = 500 \cdot e^{0.02 \cdot 0} = 500\).

Final Answer:

The quantity \(Q(t)\) at time \(t = 0\) is approximately 500, rounded to 0 decimal places.

Step 1: Identify the Initial Quantity

The initial quantity \(Q_0\) given is 500.

Step 2: Determine the Growth (or Decay) Constant

The growth (or decay) constant \(k\) given is 0.02.

Step 3: Calculate the Quantity \(Q(t)\)

Using the formula \(Q(t) = Q_0 e^{kt}\), where \(t = 10\), we calculate: \(Q(10) = 500 \cdot e^{0.02 \cdot 10} = 611\).

Final Answer:

The quantity \(Q(t)\) at time \(t = 10\) is approximately 611, rounded to 0 decimal places.

Step 1: Identify the Initial Quantity

The initial quantity \(Q_0\) given is 500.

Step 2: Determine the Growth (or Decay) Constant

The growth (or decay) constant \(k\) given is 0.02.

Step 3: Calculate the Quantity \(Q(t)\)

Using the formula \(Q(t) = Q_0 e^{kt}\), where \(t = 20\), we calculate: \(Q(20) = 500 \cdot e^{0.02 \cdot 20} = 746\).

Final Answer:

The quantity \(Q(t)\) at time \(t = 20\) is approximately 746, rounded to 0 decimal places.

Step 1: Identify the Initial Quantity

The initial quantity \(Q_0\) given is 500.

Step 2: Determine the Growth (or Decay) Constant
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