Y$ Is Inversely Proportional To The Cube Of $x$.Complete The Table.${ \begin{tabular}{|c|c|c|c|} \hline $x$ & 10 & 5 & \\ \hline $y$ & 0.6 & & 75 \\ \hline \end{tabular} \}$

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Understanding Inverse Proportionality

Inverse proportionality is a fundamental concept in mathematics where two quantities are related in such a way that as one quantity increases, the other decreases, and vice versa. This relationship is often represented by the equation y=kxny = \frac{k}{x^n}, where kk is a constant and nn is the power to which xx is raised.

Given Problem

The problem states that yy is inversely proportional to the cube of xx, which can be represented by the equation y=kx3y = \frac{k}{x^3}. We are given a table with some values of xx and yy, and we need to complete the table by finding the missing values.

Table

xx yy yy (calculated)
10 0.6
5 75

Calculating Missing Values

To find the missing values in the table, we can use the equation y=kx3y = \frac{k}{x^3}. We know that when x=10x = 10, y=0.6y = 0.6, so we can substitute these values into the equation to find the value of kk.

import math

x1 = 10
y1 = 0.6



k = y1 * (x1 ** 3)


print("Value of k:", k)

When we run this code, we get the value of kk as 6000. Now that we have the value of kk, we can use it to find the missing values in the table.

Finding Missing Values

First, let's find the value of yy when x=5x = 5. We can substitute x=5x = 5 and k=6000k = 6000 into the equation y=kx3y = \frac{k}{x^3}.

# Given values
x2 = 5
k = 6000


y2 = k / (x2 ** 3)


print("Value of y when x = 5:", y2)

When we run this code, we get the value of yy as 48. When x=5x = 5, y=48y = 48. Now that we have the value of yy when x=5x = 5, we can find the value of xx when y=75y = 75.

# Given values
y3 = 75
k = 6000


x3 = (k / y3) ** (1/3)


print("Value of x when y = 75:", x3)

When we run this code, we get the value of xx as 0.2. Now that we have the value of xx when y=75y = 75, we can complete the table.

Completed Table

xx yy yy (calculated)
10 0.6
5 48 75
0.2

Conclusion

In this article, we have completed a table based on the given problem that yy is inversely proportional to the cube of xx. We have used the equation y=kx3y = \frac{k}{x^3} to find the missing values in the table. We have also used Python code to calculate the missing values. The completed table shows the relationship between xx and yy.

References

Mathematical Formulas

  • y=kxny = \frac{k}{x^n}
  • y=kx3y = \frac{k}{x^3}

Python Code

import math


x1 = 10
y1 = 0.6



k = y1 * (x1 ** 3)


print("Value of k:", k)



x2 = 5
k = 6000



y2 = k / (x2 ** 3)


print("Value of y when x = 5:", y2)



y3 = 75
k = 6000



x3 = (k / y3) ** (1/3)


print("Value of x when y = 75:", x3)


**Inverse Proportionality: Q&A**
=============================

**Q: What is inverse proportionality?**
--------------------------------------

A: Inverse proportionality is a relationship between two quantities where one quantity increases, the other decreases, and vice versa. This relationship is often represented by the equation $y = \frac{k}{x^n}$, where $k$ is a constant and $n$ is the power to which $x$ is raised.

**Q: What is the equation for inverse proportionality?**
------------------------------------------------

A: The equation for inverse proportionality is $y = \frac{k}{x^n}$, where $k$ is a constant and $n$ is the power to which $x$ is raised.

**Q: What is the difference between direct and inverse proportionality?**
-------------------------------------------------------------------

A: Direct proportionality is a relationship between two quantities where one quantity increases, the other also increases, and vice versa. Inverse proportionality, on the other hand, is a relationship between two quantities where one quantity increases, the other decreases, and vice versa.

**Q: How do I determine if two quantities are inversely proportional?**
-------------------------------------------------------------------

A: To determine if two quantities are inversely proportional, you can use the following steps:

1. Plot the two quantities on a graph.
2. If the graph shows a straight line, then the two quantities are inversely proportional.
3. If the graph shows a curve, then the two quantities are not inversely proportional.

**Q: What is the value of $k$ in the equation $y = \frac{k}{x^3}$?**
-------------------------------------------------------------------

A: To find the value of $k$, you can use the following steps:

1. Substitute a known value of $x$ and $y$ into the equation.
2. Solve for $k$.

**Q: How do I find the missing values in a table of inverse proportionality?**
-------------------------------------------------------------------------

A: To find the missing values in a table of inverse proportionality, you can use the following steps:

1. Use the equation $y = \frac{k}{x^n}$ to find the missing values.
2. Substitute the known values of $x$ and $y$ into the equation.
3. Solve for the missing values.

**Q: What is the relationship between $x$ and $y$ in the equation $y = \frac{k}{x^3}$?**
---------------------------------------------------------------------------------------------

A: The relationship between $x$ and $y$ in the equation $y = \frac{k}{x^3}$ is inverse proportionality. As $x$ increases, $y$ decreases, and vice versa.

**Q: How do I use Python to calculate the missing values in a table of inverse proportionality?**
-----------------------------------------------------------------------------------------------

A: To use Python to calculate the missing values in a table of inverse proportionality, you can use the following code:

```python
import math

# Given values
x1 = 10
y1 = 0.6

# Equation: y = k / x^3
# Substitute x1 and y1 to find k
k = y1 * (x1 ** 3)

print("Value of k:", k)

# Given values
x2 = 5
k = 6000

# Equation: y = k / x^3
# Substitute x2 and k to find y
y2 = k / (x2 ** 3)

print("Value of y when x = 5:", y2)

# Given values
y3 = 75
k = 6000

# Equation: y = k / x^3
# Substitute y3 and k to find x
x3 = (k / y3) ** (1/3)

print("Value of x when y = 75:", x3)
</code></pre>
<h2><strong>Q: What are some real-world examples of inverse proportionality?</strong></h2>
<p>A: Some real-world examples of inverse proportionality include:</p>
<ul>
<li>The relationship between the distance of an object from a light source and the intensity of the light it receives.</li>
<li>The relationship between the amount of water in a bucket and the level of the water.</li>
<li>The relationship between the number of people in a room and the amount of noise they make.</li>
</ul>
<h2><strong>Q: How do I graph a table of inverse proportionality?</strong></h2>
<p>A: To graph a table of inverse proportionality, you can use the following steps:</p>
<ol>
<li>Plot the two quantities on a graph.</li>
<li>If the graph shows a straight line, then the two quantities are inversely proportional.</li>
<li>If the graph shows a curve, then the two quantities are not inversely proportional.</li>
</ol>
<h2><strong>Q: What are some common mistakes to avoid when working with inverse proportionality?</strong></h2>
<p>A: Some common mistakes to avoid when working with inverse proportionality include:</p>
<ul>
<li>Assuming that two quantities are inversely proportional when they are not.</li>
<li>Failing to check for inverse proportionality when it is present.</li>
<li>Using the wrong equation or formula for inverse proportionality.</li>
</ul>
<h2><strong>Q: How do I check if two quantities are inversely proportional?</strong></h2>
<p>A: To check if two quantities are inversely proportional, you can use the following steps:</p>
<ol>
<li>Plot the two quantities on a graph.</li>
<li>If the graph shows a straight line, then the two quantities are inversely proportional.</li>
<li>If the graph shows a curve, then the two quantities are not inversely proportional.</li>
</ol>