https://file.aiquickdraw.com/mathbot/file/1726484774725xdmgkd3x.jpg

We are given a table with two values of \( x \) and their corresponding \( y \) values, along with the information that \( y \) is proportional to \( x^2 \), i.e., \( y \propto x^2 \). This implies that:

\[
y = kx^2
\]

where \( k \) is the constant of proportionality.

The values of \( x \) and \( y \) from the table are:

\[
x = 5, \quad y = 400
\]
\[
x = 6, \quad y = 576
\]

### Step 1: Find the constant \( k \)
Using the first pair of values \( x = 5 \) and \( y = 400 \), substitute these into the equation \( y = kx^2 \):

\[
400 = k(5)^2
\]
\[
400 = k(25)
\]
\[
k = \frac{400}{25} = 16
\]

### Step 2: Write the equation for \( y \)
Now that we know \( k = 16 \), the equation for \( y \) in terms of \( x \) becomes:

\[
y = 16x^2
\]

### Verification
We can verify this by substituting \( x = 6 \) into the formula to check if it gives \( y = 576 \):

\[
y = 16(6)^2 = 16 \times 36 = 576
\]

This matches the given value of \( y \), so the formula is correct.

### Final Answer:
\[
y = 16x^2
\]

Would you like further clarification on any part of this solution?

Here are five related questions for further exploration:
1. How would the formula change if \( y \) were proportional to \( x^3 \)?
2. What happens to the value of \( y \) if \( x \) is doubled?
3. How can we determine the constant \( k \) if different values of \( x \) and \( y \) are provided?
4. What is the value of \( y \) when \( x = 7 \) using the derived formula?
5. Can you derive the proportionality constant \( k \) from a different set of points?

**Tip:** When dealing with proportionality problems, identifying the constant of proportionality is key, and it usually requires using known values of variables to solve for it.

Write a formula for y in terms of x, given the table values and y ∝ x².

Learn how to derive the formula y = kx² from given x and y values, where y is proportional to x². The step-by-step process explains how to calculate the constant of proportionality (k) using x = 5, y = 400 and x = 6, y = 576. Ideal for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation