Problem 455 - Precalculus

Question

For the following exercises, refer to Table 4.26. $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} \\\ \hline f(x) & {1125} & {1495} & {2310} & {3294} & {4650} & {6361} \\\ \hline\end{array}$$ Use the intersect feature to find the value of \(x\) for which \(f(x)=4000\) .

Step-by-Step Solution

Answer

The value of \( x \) for which \( f(x) = 4000 \) is approximately 4.52.

Step 1: Understanding the Problem

We need to find the value of \( x \) for which \( f(x) = 4000 \). The given table provides values of \( x \) and corresponding values of \( f(x) \). We are to use the intersect feature, which typically involves finding where two functions intersect. In this case, we consider the horizontal line \( y = 4000 \) and the function defined by the table.

Step 2: Interpolating Between Known Values

Looking at the table, \( f(x) \) jumps from 3294 at \( x = 4 \) to 4650 at \( x = 5 \). Therefore, 4000 must fall between these two \( x \)-values (4 and 5). We will interpolate to estimate \( x \) for which \( f(x) = 4000 \).

Step 3: Linear Interpolation Formula

For linear interpolation between the points \((4, 3294)\) and \((5, 4650)\), use the formula:\[x = x_1 + \frac{(y - y_1)(x_2 - x_1)}{y_2 - y_1}\]where \( (x_1, y_1) = (4, 3294) \) and \( (x_2, y_2) = (5, 4650) \). Substitute \( y = 4000 \) into the equation.

Step 4: Calculate the Interpolated x Value

Substitute into the interpolation formula:\[x = 4 + \frac{(4000 - 3294)(5 - 4)}{4650 - 3294}\]Calculate:\[x = 4 + \frac{706}{1356}\]\[x \approx 4.52\]. This is the estimated value of \( x \) for which \( f(x) = 4000 \).

Key Concepts

The Interpolation Formula

To understand how to apply the interpolation formula, start by knowing that this tool estimates unknown values between known data points. Linear interpolation provides an estimated result by assuming that the change between two data points is linear or uniform.

In this exercise, you have two key points:

• One at \((x_1, y_1) = (4, 3294)\)
• Another at \((x_2, y_2) = (5, 4650)\)

We aim to find an \( x \) that corresponds to a specific \( y \), here \( 4000 \). The formula is:\[ x = x_1 + \frac{(y - y_1)(x_2 - x_1)}{y_2 - y_1}\]The formula inputs the known values from the data table to output a precise \( x \). Linear interpolation simplifies the task of finding values that fall between known data points.

Understanding Table Values

Tables are a vital part of understanding structured data. They represent relationships between variables—in this case, \( x \) and \( f(x) \). The table given in the exercise presents six pairs:

• \((1, 1125)\)
• \((2, 1495)\)
• \((3, 2310)\)
• \((4, 3294)\)
• \((5, 4650)\)
• \((6, 6361)\)

Understanding data in tabular form allows us to make calculations effortless. It identifies where the desired output (\( f(x) = 4000 \)) might fall between provided inputs (\( x \) values). Recognizing the jump from \( 3294 \) to \( 4650 \) helps decide that an interpolation is possible between \( x = 4 \) and \( x = 5 \).

Tables simplify the analysis, acting as an organized summary for quick interpretation.

The Concept of Function Intersection

Function intersection is a critical idea when determining how two different functions relate. The term 'intersect' implies the point where two graphs meet. For this problem, the goal was to find when the horizontal line \( y = 4000 \) cuts across the curve derived from the table values of \( f(x) \).

This problem specifically considers only one function derived from table data and another constant, \( f(x) = 4000 \). Thus, finding the intersection involves estimating the position \( x \) where the given formula for \( f(x) \) equals \( 4000 \). By setting up the problem as an intersection, it helps visualize how the numerical value is found using both graphical ideas and numerical processes.

How to Estimate the x-Value

Estimating an \( x \)-value, as demonstrated in this exercise, hinges upon using interpolation to predict values that aren’t immediately visible. Here's how it works:

Using the formula for interpolation, substitute known \( y_1 \), \( y_2 \), and desired \( y = 4000 \) into:\[x = 4 + \frac{(4000 - 3294)(5 - 4)}{4650 - 3294}\]Once inserted, calculate each step:

• Find the difference: \( 4000 - 3294 = 706 \)
• Use the fraction built around this difference: \( \frac{706}{1356} \approx 0.52 \)
• Result by adding to \( x_1 \): \( 4 + 0.52 = 4.52 \)

Thus, an estimated \( x \) value of approximately \( 4.52 \), meaning \( x \) is slightly more than 4 yet less than 5 when \( f(x) \) equals 4000. This method gives us a nuanced approach to finding numbers beyond simple measurement.