Math Problem Statement

y is inversely proportional to x^2. When x = 4, y = 2. Find y when x = 1/2.

Solution

To solve this problem, let's go through the steps.

Step 1: Set up the relationship

Since yy is inversely proportional to x2x^2, we can express this as: y=kx2y = \frac{k}{x^2} where kk is a constant.

Step 2: Find the constant kk

We know that when x=4x = 4, y=2y = 2. Substitute these values into the equation to find kk: 2=k422 = \frac{k}{4^2} 2=k162 = \frac{k}{16} Multiply both sides by 16: k=32k = 32

Step 3: Rewrite the equation with the constant

Now that we know k=32k = 32, the equation becomes: y=32x2y = \frac{32}{x^2}

Step 4: Find yy when x=12x = \frac{1}{2}

Substitute x=12x = \frac{1}{2} into the equation: y=32(12)2y = \frac{32}{\left( \frac{1}{2} \right)^2} Calculate (12)2\left( \frac{1}{2} \right)^2: (12)2=14\left( \frac{1}{2} \right)^2 = \frac{1}{4} So the equation becomes: y=3214y = \frac{32}{\frac{1}{4}} Dividing by 14\frac{1}{4} is the same as multiplying by 4: y=32×4=128y = 32 \times 4 = 128

Final Answer

y=128y = 128

Would you like a further breakdown of these steps, or do you have any questions?


Here are five related questions that might interest you:

  1. How does inverse proportionality differ from direct proportionality?
  2. What happens to yy if xx is doubled in this inverse relationship?
  3. How can we interpret inverse proportionality graphically?
  4. How would the relationship change if yy were proportional to xx instead of x2x^2?
  5. Can this approach be used for other powers of xx, such as x3x^3?

Tip: Remember, when dealing with inverse proportionality, as xx increases, yy decreases, and vice versa.

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Math Problem Analysis

Mathematical Concepts

Inverse Proportion
Exponents
Algebra

Formulas

y = k / x^2

Theorems

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Suitable Grade Level

Grades 8-10