x is inversely proportional to the cube of y. its known that x = 9 for a particular value of y. when the value of y is increased by 100%, find the value of x, find the percentage decrease in the value of x

Since \( x \) is inversely proportional to the cube of \( y \), we can write this relationship as:
\[ x \propto \frac{1}{y^3} \]
This can be expressed with a constant \( k \):
\[ x = \frac{k}{y^3} \]

Given that \( x = 9 \) for a particular value of \( y \), let's denote this specific value of \( y \) as \( y_1 \). Therefore:
\[ 9 = \frac{k}{y_1^3} \]

From this, we can solve for \( k \):
\[ k = 9 y_1^3 \]

Now, we are asked to find the value of \( x \) when \( y \) is increased by 100%. If \( y \) is increased by 100%, the new value of \( y \) becomes:
\[ y_2 = 2y_1 \]

Using the relationship \( x = \frac{k}{y^3} \) with the new value of \( y \):
\[ x_2 = \frac{k}{(2y_1)^3} \]
\[ x_2 = \frac{9 y_1^3}{8 y_1^3} \]
\[ x_2 = \frac{9}{8} \]

Thus, the new value of \( x \) is \( \frac{9}{8} \).

Next, we calculate the percentage decrease in the value of \( x \) from the original value (9) to the new value (\(\frac{9}{8}\)):

1. Find the difference in values:
\[ \Delta x = 9 - \frac{9}{8} \]
\[ \Delta x = 9 - 1.125 \]
\[ \Delta x = 7.875 \]

2. Calculate the percentage decrease:
\[ \text{Percentage decrease} = \left( \frac{\Delta x}{\text{original } x} \right) \times 100\% \]
\[ \text{Percentage decrease} = \left( \frac{7.875}{9} \right) \times 100\% \]
\[ \text{Percentage decrease} \approx 87.5\% \]

So, when \( y \) is increased by 100%, the value of \( x \) decreases by approximately 87.5%.

Would you like more details or have any questions?

Here are some related questions you might be interested in:
1. How do you solve problems involving direct proportionality?
2. Can you explain how to solve problems involving joint variation?
3. What are some real-life applications of inverse proportionality?
4. How do you derive the constant of proportionality in different scenarios?
5. What is the difference between linear and non-linear proportional relationships?

**Tip:** When dealing with proportional relationships, always identify the type of proportionality (direct or inverse) and the variables involved to set up the correct equation.

Explore how to solve problems involving inverse proportionality where x is inversely proportional to the cube of y. Includes a detailed step-by-step solution and calculation of percentage decrease.

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