https://file.aiquickdraw.com/mathbot/file/1728290222926soccu9xv.jpg

To find the value of \( y(2.9) \) from the table provided, we can use **interpolation**, which estimates the value of a function between known data points.

The table provides us with the following pairs of \( x \) and \( y \):

\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
y & 3 & 3 & 19 & 69 & 171 & 343 \\
\hline
\end{array}
\]

Since \( 2.9 \) lies between \( x = 2 \) and \( x = 3 \), we will use **linear interpolation** to estimate \( y(2.9) \).

The formula for linear interpolation between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[
y(x) = y_1 + \frac{(y_2 - y_1)}{(x_2 - x_1)} (x - x_1)
\]

### Applying the values:
- \( x_1 = 2 \), \( y_1 = 19 \)
- \( x_2 = 3 \), \( y_2 = 69 \)
- \( x = 2.9 \)

Using the formula:

\[
y(2.9) = 19 + \frac{(69 - 19)}{(3 - 2)}(2.9 - 2)
\]

\[
y(2.9) = 19 + 50 \times 0.9
\]

\[
y(2.9) = 19 + 45 = 64
\]

### Therefore, \( y(2.9) \approx 64 \).

Would you like more details on interpolation, or have any questions?

Here are 5 related questions to deepen your understanding:
1. What is the general difference between linear and polynomial interpolation?
2. How would the result change if quadratic interpolation was used instead of linear?
3. Can you perform interpolation for values outside the given range (e.g., extrapolation)?
4. How is interpolation used in real-world applications, such as data analysis or computer graphics?
5. What are the limitations of using interpolation with non-linear data?

**Tip**: Always check the interval size between known data points—if they change significantly, linear interpolation might not be the best approximation.

Find the value of y(2.9) from the following table, where the table provides the values of x: [0, 1, 2, 3, 4, 5] and corresponding values of y: [3, 3, 19, 69, 171, 343].

Learn how to estimate the value of y(2.9) from the given table using linear interpolation. This problem explores interpolation techniques and provides a detailed calculation for estimating y between known data points. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation