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Chapter 3: Problem 2

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, answer the following questions: a. What is the decay constant? b. What quantity is present initially? c. Complete the following table of values:

Short Answer

Expert verified
a. The decay constant is \(\lambda = 0.06\). b. The initial quantity present is \(Q_0 = 2000\). c. The completed table of values is: | \(t\) (Years) | \(Q(t)\) | |-------------|-----------| | 0 | 2000 | | 1 | 1248.69 | | 2 | 778.80 | | 3 | 485.18 | | 4 | 302.35 | | 5 | 188.37 |

Step by step solution

01

The given exponential decay function is in the form: $$ Q(t) = Q_0 e^{(-\lambda t)} $$ where \(Q_0\) is the initial quantity, \(\lambda\) is the decay constant, and \(t\) is time in years. #Step 2: Find the decay constant#

Comparing the given function \(Q(t) = 2000 e^{-0.06t}\) with the general formula, we can identify the decay constant \(\lambda\) as -0.06 (keeping in mind decay constant is positive, and the formula contains a negative sign). So, the decay constant is \(\lambda = 0.06\). #Step 3: Find the initial quantity present#
02

Again, by comparing the given function with the general formula, we can identify the initial quantity \(Q_0\) as 2000. So, the initial quantity present is \(Q_0 = 2000\). #Step 4: Complete the table of values#

We are asked to complete a table of values for the given function. We will list the \(t\) values (in years) given and calculate the corresponding \(Q(t)\) values using the given function. | \(t\) (Years) | \(Q(t)\) | |-------------|-----------| | 0 | | | 1 | | | 2 | | | 3 | | | 4 | | | 5 | | For each \(t\) value, substitute it into the given function: \(Q(t) = 2000 e^{-0.06t}\) | \(t\) (Years) | \(Q(t)\) | |-------------|-----------------| | 0 | \(2000 e^{-0.06*0}\) | | 1 | \(2000 e^{-0.06*1}\) | | 2 | \(2000 e^{-0.06*2}\) | | 3 | \(2000 e^{-0.06*3}\) | | 4 | \(2000 e^{-0.06*4}\) | | 5 | \(2000 e^{-0.06*5}\) | Now we calculate and fill in the \(Q(t)\) values: | \(t\) (Years) | \(Q(t)\) | |-------------|-----------| | 0 | 2000 | | 1 | 1248.69 | | 2 | 778.80 | | 3 | 485.18 | | 4 | 302.35 | | 5 | 188.37 | So, the completed table of values for the given function is as shown above.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decay Constant
When discussing exponential decay, a key concept to understand is the decay constant, often represented by \(\lambda\). This constant is vital because it determines the rate at which a quantity diminishes over time.

Generally speaking, in the function \(Q(t) = Q_0 e^{-\lambda t}\), the negative sign indicates that the quantity is decreasing, and the value of \(\lambda\) tells us how fast it's happening. A higher decay constant suggests a faster rate of decay.

For example, in the given function \(Q(t) = 2000 e^{-0.06t}\), by comparing it with the general formula, \(\lambda\) is identified as \(0.06\) per year. It means that for each year that passes, the quantity will reduce by approximately 6% of its value at the beginning of that year. Remember, although \(\lambda\) appears with a negative sign in the formula, it is a positive value indicating the decay's natural behavior.
Initial Quantity
The term 'initial quantity' refers to the amount of a substance or the size of a quantity at the beginning of the period being examined, before any decay has occurred. It's denoted as \(Q_0\) in the exponential decay function \(Q(t) = Q_0 e^{-\lambda t}\).

The initial quantity is an essential baseline because it allows us to predict future values using the exponential decay model. In the context of the exercise, the initial quantity is found by examining the value when \(t = 0\). Following the function \(Q(t) = 2000 e^{-0.06t}\), we can see that \(Q_0 = 2000\). This means that at time \(t = 0\), the quantity in question was 2000 units. From this starting point, we can utilize the exponential decay function to estimate the quantity at any future point in time.
Exponential Decay Function
An exponential decay function represents how a quantity decreases over time at a rate proportional to its current value. It is often expressed as \(Q(t) = Q_0 e^{-\lambda t}\) where \(Q(t)\) is the quantity at time \(t\), \(Q_0\) is the initial quantity, and \(e\) is the base of the natural logarithm, approximately equal to 2.71828.

This mathematics concept is observed in a variety of natural phenomena, including radioactive decay and the cooling of an object. For instance, in our exercise, the value of \(Q(t)\) for different \(t\) values was calculated to create a table of values, demonstrating how the quantity decreases over time. As \(t\) increases, \(e^{-\lambda t}\) decreases, which consequently reduces \(Q(t)\), exemplifying the very nature of exponential decay.

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Most popular questions from this chapter

On the Richter scale, the magnitude \(R\) of an earthquake is given by the formula $$ R=\log \frac{I}{I_{0}} $$ where \(I\) is the intensity of the earthquake being measured and \(I_{0}\) is the standard reference intensity. a. Express the intensity \(I\) of an earthquake of magnitude \(R=5\) in terms of the standard intensity \(I_{0}\). b. Express the intensity \(I\) of an earthquake of magnitude \(R=8\) in terms of the standard intensity \(I_{0}\). How many times greater is the intensity of an earthquake of magnitude 8 than one of magnitude \(5 ?\) c. In modern times, the greatest loss of life attributable to an earthquake occurred in eastern China in 1976 . Known as the Tangshan earthquake, it registered \(8.2\) on the Richter scale. How does the intensity of this earthquake compare with the intensity of an earthquake of magnitude \(R=5 ?\)

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