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Problem 1

# Given that a quantity $$Q(t)$$ is described by the exponential growth function $$Q(t)=400 e^{\mathrm{a} .05 t}$$ where $$t$$ is measured in minutes, answer the following questions: a. What is the growth constant? b. What quantity is present initially? c. Complete the following table of values:

Expert verified
a. The growth constant is $$0.05\mathrm{a}$$. b. The initial quantity present is 400 units. c. The completed table of values is: | Time (t) | Quantity (Q) | |:--------------:|:----------------------------:| | 0 | 400 | | 5 | $$400e^{0.25\mathrm{a}}$$ | | 10 | $$400e^{0.5\mathrm{a}}$$ | | 15 | $$400e^{0.75\mathrm{a}}$$ |
See the step by step solution

## Step 1: Identify the growth constant

The growth constant is the coefficient of $$t$$ in the given exponent of the function: $$\mathrm{a} \cdot 0.05$$. Therefore, the growth constant is $$0.05\mathrm{a}$$.

## Step 2: Determine the initial value

In order to determine the initial quantity present in the system, we must evaluate $$Q(t)$$ when $$t=0$$. That is: $$Q(0) = 400e^{\mathrm{a} \cdot 0.05\cdot 0} = 400e^0 = 400$$ So, initially, there are 400 units of the quantity present.

## Step 3: Complete the table of values

We will now find the value of the function at different times $$t$$. We can organize these values in a table as shown below: | Time (t) | Quantity (Q) | |:--------------:|:----------------:| | 0 | 400 | | 5 | $$400e^{0.05\cdot 5\cdot \mathrm{a}}$$ | | 10 | $$400e^{0.05\cdot 10\cdot \mathrm{a}}$$ | | 15 | $$400e^{0.05\cdot 15\cdot \mathrm{a}}$$ | Using the given formula, we can calculate the value of $$Q(t)$$ for the given time values: For $$t=5$$ minutes: $$Q(5) = 400e^{0.05\cdot 5\cdot \mathrm{a}} = 400e^{0.25\mathrm{a}}$$ For $$t=10$$ minutes: $$Q(10) = 400e^{0.05\cdot 10\cdot \mathrm{a}} = 400e^{0.5\mathrm{a}}$$ For $$t=15$$ minutes: $$Q(15) = 400e^{0.05\cdot 15\cdot \mathrm{a}} = 400e^{0.75\mathrm{a}}$$ The completed table of values is as follows: | Time (t) | Quantity (Q) | |:--------------:|:----------------------------:| | 0 | 400 | | 5 | $$400e^{0.25\mathrm{a}}$$ | | 10 | $$400e^{0.5\mathrm{a}}$$ | | 15 | $$400e^{0.75\mathrm{a}}$$ |

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