Problem 1
Question
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:
Step-by-Step Solution
Verified Answer
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 | \(400e^{0.75\mathrm{a}}\) |
1Step 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 \(0.05\mathrm{a}\).
2Step 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.
3Step 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 | \(400e^{0.75\mathrm{a}}\) |
Key Concepts
Growth ConstantInitial ValueTable of Values
Growth Constant
In the world of mathematics, the concept of a 'growth constant' is crucial when discussing exponential growth functions. When we look at an exponential function like the one given in our exercise, \( Q(t) = 400 \, e^{0.05a \, t} \) where \( t \) is time, the growth constant is the factor that determines how quickly the quantity increases over time. It's akin to the 'speed' of growth.
In this specific function, the growth constant is represented by \( 0.05a \). If we think of 'a' as a sort of tuning knob, adjusting its value will impact our growth constant and thus, the rate at which our quantity \( Q \) grows. A larger 'a' means a steeper increase, and a smaller 'a' equates to a more gradual rise. Understanding the growth constant is essential for predicting future values and interpreting the behavior of the function.
In this specific function, the growth constant is represented by \( 0.05a \). If we think of 'a' as a sort of tuning knob, adjusting its value will impact our growth constant and thus, the rate at which our quantity \( Q \) grows. A larger 'a' means a steeper increase, and a smaller 'a' equates to a more gradual rise. Understanding the growth constant is essential for predicting future values and interpreting the behavior of the function.
Initial Value
Every journey has a starting point, and in the case of exponential functions, this is what we call the 'initial value'. It's the value of the function at the start of our observations, when \( t = 0 \). To find it, we simply input 0 for \( t \) in our function, making the exponent equal to zero.
The power of e to zero is always 1, hence any number times 1 remains unchanged, which is why \( Q(0) = 400e^{0} = 400 \). The initial value is crucial since it's the baseline from which we measure all subsequent growth. In real-world applications, knowing the initial quantity gives context to the numbers we see later on. It tells us, for example, how many bacteria we started with in a petri dish, or the initial amount of money in a bank account that's earning compound interest.
The power of e to zero is always 1, hence any number times 1 remains unchanged, which is why \( Q(0) = 400e^{0} = 400 \). The initial value is crucial since it's the baseline from which we measure all subsequent growth. In real-world applications, knowing the initial quantity gives context to the numbers we see later on. It tells us, for example, how many bacteria we started with in a petri dish, or the initial amount of money in a bank account that's earning compound interest.
Table of Values
A 'table of values' is a systematic way to display how the quantity we're looking at changes over time—it's a snapshot of the function's behavior. To fill in this table for our exponential growth function, we calculate the function's values at selected intervals of time. This approach transforms the abstract equation into tangible numbers that we can interpret more straightforwardly.
As we calculate values for \( t = 5 \) and so on, the terms \( 400e^{0.25a} \), \( 400e^{0.5a} \), and \( 400e^{0.75a} \) fill our table and reveal the exponential trend. Constructing and analyzing such a table not only aids in understanding the function's trajectory but also enhances comprehension of concepts like growth constant and initial value, connecting the numeric output to the function's parameters.
As we calculate values for \( t = 5 \) and so on, the terms \( 400e^{0.25a} \), \( 400e^{0.5a} \), and \( 400e^{0.75a} \) fill our table and reveal the exponential trend. Constructing and analyzing such a table not only aids in understanding the function's trajectory but also enhances comprehension of concepts like growth constant and initial value, connecting the numeric output to the function's parameters.
Other exercises in this chapter
Problem 1
Express each equation in logarithmic form. $$2^{6}=64$$
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Evaluate the expression. a. \(4^{-3} \cdot 4^{5}\) b. \(3^{-3}+3^{6}\)
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Given that a quantity \(Q(t)\) exhibiting exponential decay is described by the function $$ Q(t)=2000 e^{-0.06 \mathrm{~s}} $$ where \(t\) is measured in years,
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