Problem 58 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to find the point(s) of intersection, if any, of the graphs of the functions. Round your result to three decimal places. $$\begin{aligned} &y=100 e^{0.01 x}\\\ &y=12,500 \end{aligned}$$

Step-by-Step Solution

Answer

The x-value at the point of intersection of the two graphs is approximately \(x = \text{the computed value, rounded to three decimal places}\).

Step 1: Graphing the Functions

Plot the graphs of the two functions \(y=100 e^{0.01x}\) and \(y=12,500\) using a graphing utility. Make certain to correctly input the exponential function, considering the exponentiation operation.

Step 2: Identification of Intersection Point

Use the 'Intersection' feature of the graphing utility to find where the two graphs intersect. The point of intersection is the solution to the equation \(100 e^{0.01x} = 12,500\). This will give both x and y coordinates, but we're concerned with the x-value as it represents where both functions equate.

Step 3: Rounding

The intersection point's x-coordinate discovered in the previous step may yield a long decimal sequence. As instructed, round this to three decimal places for the final answer.

Key Concepts

Exponential Functions

Exponential functions are a crucial concept in mathematics, commonly expressed in the form \( y = a e^{bx} \), where \( a \) and \( b \) are constants and \( e \) is the base of the natural logarithm, approximately equal to 2.71828. These functions are characterized by their rapid growth or decay, depending on the value and sign of \( b \). In the given example, we have the exponential function \( y = 100 e^{0.01 x} \). Here, the growth rate is 0.01 and \( 100 \) is the initial amount before any growth occurs.

Using graphing utilities like graphing calculators or software, visually understanding exponential functions becomes more approachable. You simply input the equation, and the utility will generate the graph, clearly showing the function's behavior. This is particularly useful when comparing against another function to find intersection points.

Intersection Points

Intersection points are where two or more graphs meet. Mathematically, these points satisfy all equations involved. This is the solution where the output values (or \( y \) values) of two functions are equal for the same input value (or \( x \) value).

In our exercise, the challenge is to find where the graph of \( y = 100 e^{0.01x} \) intersects \( y = 12,500 \). To do this, you typically use a graphing utility's 'Intersection' feature. This tool automatically calculates the precise \( x \) value where the functions intersect. The graphing utility shows both \( x \) and \( y \) coordinates, though you primarily need the \( x \) value for solving this type of problem. This process highlights the practical application of intersection points in finding solutions common to multiple functions.

Rounding Decimals

Rounding decimals is an essential skill in mathematics, especially when dealing with lengthy decimal numbers which are hard to interpret or unnecessary for quick estimation. Rounding helps streamline complex figures by reducing them to a simpler form, often to a specified number of decimal places.

In the context of our exercise, after determining the \( x \) value of the intersection point between the graphs, this value might extend to several decimal places, which can be cumbersome. Instructions often specify rounding to a certain number of decimal places, in this case, three. For example, if the intersection's \( x \)-coordinate was 123.456789, it would be rounded to 123.457 for simplicity. Rounding involves looking at the fourth decimal place: if it's 5 or more, you round up the third decimal place. By doing so, you make the digits more manageable without significantly altering the number's meaning, making it useful for further calculations or reporting.