Problem 36 - Algebra 1: Concepts and Skills

Question

Use the following information. The air intake b (in liters per minute) of a cyclist on a racing bike can be modeled by \(b=6.37(1.11)^{s},\) where s is the speed of the bike (in miles per hour). Graph the exponential growth model.

Step-by-Step Solution

Answer

The graph will show the relationship where as the speed (s) increases, the air intake (b) also increases, following an exponential growth rate. The graph starts at a point (0,6.37) - indicating that even at 0 speed, there is some air intake, and as the cyclist's speed increases, the air intake rate significantly increases.

Step 1: Understanding the relationship

The given equation \(b=6.37(1.11)^{s}\) is an exponential function, where '6.37' is the initial value (when s=0), '1.11' is the base which indicates the growth rate, and 's' is the exponent representing the speed at which the cyclist is riding.

Step 2: Set up for graphing

Before plotting the graph, the range of 's' values should be selected. For an easy understanding, we can consider the values of 's' from 0 to 10 with intervals of 1 mile per hour. Then, for each 's' value, substitute it into the function to get the corresponding 'b' values.

Step 3: Plot on the graph

Start plotting the points on a Cartesian coordinate plane, where 's' is on the x-axis while 'b' is on the y-axis. After plotting the points, sketch a curve through them to complete the exponential growth model graph.

Key Concepts

Graphing Exponential Functions

Graphing exponential functions like the one given for the cyclist's air intake can help us visualize how the function behaves. The equation provided, \(b=6.37(1.11)^{s}\), is a classic example of an exponential growth function. When graphing, the key steps involve:

• Identifying the initial value, which is 6.37 in this case. This value represents the air intake in liters per minute when the speed \(s\) is 0.
• Understanding the base 1.11, which reflects the growth factor. It shows how much the air intake increases for every 1 mph increase in speed.

To start graphing, first calculate several points by substituting different values of \(s\) into the function. This will provide a set of \(b\) values. Once these points are calculated, plot them, and draw a smooth curve that nicely connects the points instead of a series of straight lines. The result will be a steepening curve that illustrates a rapid increase in air intake as speed goes up.

Understanding Growth Rate in Exponential Functions

Exponential growth depicted by functions like \(b=6.37(1.11)^{s}\) showcases a rapid increase in the output as the input increases.

In this specific model, the growth rate is determined by the base, which is 1.11. This number tells us that for every increment in speed by 1 mile per hour, the air intake increases by a factor of 1.11, or 11%.

This means:

• If the speed is 1 mph, the air intake increases by 11% of the previous amount.
• If the speed doubles, the increase is even more pronounced due to the compounding effect of exponential functions.

Exponential growth can be counter-intuitive because changes seem small initially but grow significantly as the variable increases. This is due to the compounding impact, which can be understood better by looking at the graph, where the curve becomes steeper as \(s\) increases.

The Cartesian Coordinate Plane and Its Role

The Cartesian coordinate plane is an essential tool for drawing graphs of equations like our exponential function for air intake.

It consists of two perpendicular axes:

• The x-axis (horizontal), which represents the independent variable; in our case, this is the speed \(s\) of the cyclist.
• The y-axis (vertical), which characterizes the dependent variable, the air intake \(b\).

Each point on this graph corresponds to specific values of \(s\) and \(b\). For example, when plotting our function, you calculate points like \( (0, 6.37) \), \( (1, b_1) \), and so on for each selected \(s\) value. Once the points are plotted, the visual pattern reveals how the air intake grows as the speed increases.

Having a graph on a Cartesian plane helps to easily identify trends, such as the accelerating growth rate of air intake, and provides a clear, visual representation of the function's behavior.