Problem 16 - Calculus

Question

An air-freshener starts with 30 grams and evaporates. In each of the following cases, write a formula for the quantity, \(Q\) grams, of air-freshener remaining \(t\) days after the start and sketch a graph of the function. The decrease is: (a) 2 grams a day (b) \(12 \%\) a day

Step-by-Step Solution

Answer

(a) Linear: \(Q(t) = 30 - 2t\); (b) Exponential: \(Q(t) = 30 \times (0.88)^t\).

Step 1: Understand Constant Decrease

In this problem, we have two cases. First, we need to understand that in case (a), the air-freshener decreases by 2 grams every day. This is a constant, linear decrease. We can write a formula to represent this.

Step 2: Create Formula for Case (a)

If the air-freshener starts with 30 grams and loses 2 grams per day, the formula is:\[ Q(t) = 30 - 2t \] This equation represents a linear function where the air-freshener decreases by 2 grams for each day, \(t\).

Step 3: Understand Percent Decrease

In case (b), we have a \(12\%\) daily decrease, meaning each day the air-freshener retains \(88\%\) of the previous day's amount. This is a percentage-based exponential decrease.

Step 4: Create Formula for Case (b)

The formula for an exponential decrease starting with 30 grams and decreasing by \(12\%\) daily remains:\[ Q(t) = 30 \times (0.88)^t \] This equation shows an exponential function where \(Q\) decreases by a factor of \(0.88\) each day.

Step 5: Sketch Graph for Case (a)

To sketch the graph for case (a), plot \(Q(t) = 30 - 2t\). The graph is a straight line starting at \(30\) grams when \(t = 0\) and decreasing linearly with a slope of \(-2\).

Step 6: Sketch Graph for Case (b)

For case (b), plot \(Q(t) = 30 \times (0.88)^t\). This graph starts at \(30\) grams when \(t = 0\) and curves downward, showing an exponential decrease.

Key Concepts

Linear Function

Linear functions are one of the simplest forms of mathematical relationships. They are expressed in the form of a straight line when graphed. This is because they have a constant rate of change. In the case of our air-freshener, the linear function is represented by the formula:

• \( Q(t) = 30 - 2t \)

Here, the constant rate of change is \(-2\), meaning the air-freshener loses 2 grams every day. This type of function is called linear because the decrease happens consistently over time, graphing as a straight line.
Linear functions make it easier to predict changes over time, as every additional day results in the same amount of decrease. This predictable nature is why they are often used in scenarios involving consistent, step-by-step changes.

Exponential Function

Exponential functions describe a situation where the rate of change is not constant, but rather depends on the current state. In our exercise, the air-freshener decreases by a percentage, specifically by 12% a day. The formula for this exponential decay is:

• \( Q(t) = 30 \times (0.88)^t \)

The number \(0.88\) indicates that each day, the air-freshener retains 88% of its previous amount. Unlike the linear function, where we subtracted a fixed number, here the factor of change is multiplicative, based on the existing quantity.
This type of function is significant in modeling real-world situations where the rate of change accelerates or decelerates over time, such as population growth, radioactive decay, and of course, decreasing quantities like our air-freshener.

Mathematical Modeling

Mathematical modeling involves creating mathematical expressions to represent real-world situations. In this exercise, we modeled the evaporation of an air-freshener using both linear and exponential functions.

• The linear model \( Q(t) = 30 - 2t \) shows a steady rate of evaporation.
• The exponential model \( Q(t) = 30 \times (0.88)^t \) captures the decrease as a percentage of the remaining total.

Both models serve the purpose of predicting future quantities based on current trends by applying mathematical concepts. By using these two different approaches, we can better understand and predict how the air-freshener will change over time.
Mathematical modeling is crucial in fields such as engineering, economics, and natural sciences, where predicting future behaviors based on current data is necessary.

Graphing Functions

Graphing is an essential way to visualize mathematical functions, helping us understand how quantities change over time.

• In case (a), the linear graph \( Q(t) = 30 - 2t \) is a straight downward line, steadily declining.
• For case (b), the exponential graph \( Q(t) = 30 \times (0.88)^t \) curves downward, gently at first and then more steeply.

By plotting these graphs, one can quickly see the contrast between linear and exponential functions. While the linear graph decreases at a constant rate, the exponential graph shows a diminishing rate that reduces more slowly initially but accelerates over time.
Understanding how to graph these functions enhances our ability to interpret and analyze real-world situations, making graphing an indispensable tool in the study of calculus and applied mathematics.