A Quantity \[$ B \$\] Varies Jointly With \[$ C \$\] And \[$ D \$\] And Inversely With \[$ F \$\]. When \[$ C = 4 \$\], \[$ D = 9 \$\], And \[$ F = 6 \$\], \[$ B = 18 \$\]. What Is The

Introduction

In mathematics, we often come across problems that involve relationships between variables. One such relationship is when a quantity varies jointly with two variables and inversely with another variable. In this article, we will explore this concept and use it to solve a problem.

What is Joint Variation?

Joint variation is a type of variation where a quantity varies directly with two or more variables. This means that as the values of the variables increase or decrease, the value of the quantity also increases or decreases in the same proportion. Mathematically, this can be represented as:

$$b = kcd$$

where $b$ is the quantity that varies jointly with $c$ and $d$, and $k$ is a constant of proportionality.

What is Inverse Variation?

Inverse variation is a type of variation where a quantity varies inversely with another variable. This means that as the value of the variable increases, the value of the quantity decreases, and vice versa. Mathematically, this can be represented as:

$$b = \frac{k}{f}$$

where $b$ is the quantity that varies inversely with $f$, and $k$ is a constant of proportionality.

Combining Joint and Inverse Variation

When a quantity varies jointly with two variables and inversely with another variable, we can combine the two types of variation to get a single equation. This can be represented as:

$$b = k\frac{cd}{f}$$

where $b$ is the quantity that varies jointly with $c$ and $d$, and inversely with $f$, and $k$ is a constant of proportionality.

Solving the Problem

Now that we have the equation for joint and inverse variation, we can use it to solve the problem. We are given that when $c = 4$, $d = 9$, and $f = 6$, $b = 18$. We can substitute these values into the equation to get:

$$18 = k\frac{4 \cdot 9}{6}$$

Simplifying the equation, we get:

$$18 = k\frac{36}{6}$$

$$18 = k \cdot 6$$

Dividing both sides by 6, we get:

$$3 = k$$

Now that we have found the value of $k$, we can substitute it back into the original equation to get:

$$b = 3\frac{cd}{f}$$

We can use this equation to find the value of $b$ for any given values of $c$, $d$, and $f$.

Conclusion

In this article, we have explored the concept of joint and inverse variation, and how to combine them to get a single equation. We have also used this equation to solve a problem. We have found that the value of $k$ is 3, and we have used this value to find the equation for joint and inverse variation.

Example Problems

Here are a few example problems that you can try to practice your skills:

• A quantity varies jointly with $x$ and $y$, and inversely with $z$. When $x = 2$, $y = 3$, and $z = 4$, the quantity is 12. Find the equation for joint and inverse variation.
• A quantity varies jointly with $a$ and $b$, and inversely with $c$. When $a = 5$, $b = 6$, and $c = 2$, the quantity is 30. Find the equation for joint and inverse variation.
• A quantity varies jointly with $m$ and $n$, and inversely with $p$. When $m = 3$, $n = 4$, and $p = 5$, the quantity is 24. Find the equation for joint and inverse variation.

Solutions

Here are the solutions to the example problems:

• A quantity varies jointly with $x$ and $y$, and inversely with $z$. When $x = 2$, $y = 3$, and $z = 4$, the quantity is 12. The equation for joint and inverse variation is:

$$b = k\frac{xy}{z}$$

Substituting the values of $x$, $y$, and $z$, we get:

$$12 = k\frac{2 \cdot 3}{4}$$

Simplifying the equation, we get:

$$12 = k\frac{6}{4}$$

$$12 = k \cdot 1.5$$

Dividing both sides by 1.5, we get:

$$8 = k$$

Now that we have found the value of $k$, we can substitute it back into the original equation to get:

$$b = 8\frac{xy}{z}$$

• A quantity varies jointly with $a$ and $b$, and inversely with $c$. When $a = 5$, $b = 6$, and $c = 2$, the quantity is 30. The equation for joint and inverse variation is:

$$b = k\frac{ab}{c}$$

Substituting the values of $a$, $b$, and $c$, we get:

$$30 = k\frac{5 \cdot 6}{2}$$

Simplifying the equation, we get:

$$30 = k\frac{30}{2}$$

$$30 = k \cdot 15$$

Dividing both sides by 15, we get:

$$2 = k$$

Now that we have found the value of $k$, we can substitute it back into the original equation to get:

$$b = 2\frac{ab}{c}$$

• A quantity varies jointly with $m$ and $n$, and inversely with $p$. When $m = 3$, $n = 4$, and $p = 5$, the quantity is 24. The equation for joint and inverse variation is:

$$b = k\frac{mn}{p}$$

Substituting the values of $m$, $n$, and $p$, we get:

$$24 = k\frac{3 \cdot 4}{5}$$

Simplifying the equation, we get:

$$24 = k\frac{12}{5}$$

$$24 = k \cdot 2.4$$

Dividing both sides by 2.4, we get:

$$10 = k$$

Now that we have found the value of $k$, we can substitute it back into the original equation to get:

$$b = 10\frac{mn}{p}$$

Conclusion

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Introduction

In our previous article, we explored the concept of joint and inverse variation, and how to combine them to get a single equation. We also used this equation to solve a problem. In this article, we will answer some frequently asked questions related to joint and inverse variation.

Q: What is the difference between joint and inverse variation?

A: Joint variation is a type of variation where a quantity varies directly with two or more variables. Inverse variation is a type of variation where a quantity varies inversely with another variable.

Q: How do I determine if a quantity varies jointly or inversely with another variable?

A: To determine if a quantity varies jointly or inversely with another variable, you need to analyze the relationship between the variables. If the quantity increases or decreases in the same proportion as the variables, it is a joint variation. If the quantity increases or decreases in the opposite proportion of the variable, it is an inverse variation.

Q: How do I write an equation for joint and inverse variation?

A: To write an equation for joint and inverse variation, you need to use the following formula:

$$b = k\frac{cd}{f}$$

where $b$ is the quantity that varies jointly with $c$ and $d$, and inversely with $f$, and $k$ is a constant of proportionality.

Q: How do I find the value of $k$ in the equation for joint and inverse variation?

A: To find the value of $k$ in the equation for joint and inverse variation, you need to substitute the values of $c$, $d$, and $f$ into the equation and solve for $k$.

Q: Can I use the equation for joint and inverse variation to solve problems with multiple variables?

A: Yes, you can use the equation for joint and inverse variation to solve problems with multiple variables. You just need to substitute the values of the variables into the equation and solve for the quantity.

Q: What are some real-world applications of joint and inverse variation?

A: Joint and inverse variation have many real-world applications, including:

• Physics: The motion of objects can be described using joint and inverse variation.
• Engineering: The design of systems can be described using joint and inverse variation.
• Economics: The relationship between variables such as supply and demand can be described using joint and inverse variation.

Q: How do I graph a joint and inverse variation equation?

A: To graph a joint and inverse variation equation, you need to use a graphing calculator or software. You can also use a table of values to create a graph.

Q: Can I use the equation for joint and inverse variation to solve problems with negative values?

A: Yes, you can use the equation for joint and inverse variation to solve problems with negative values. You just need to substitute the values of the variables into the equation and solve for the quantity.

Q: What are some common mistakes to avoid when working with joint and inverse variation?

A: Some common mistakes to avoid when working with joint and inverse variation include:

• Not using the correct formula for joint and inverse variation.
• Not substituting the correct values into the equation.
• Not solving for the correct variable.

Conclusion

In this article, we have answered some frequently asked questions related to joint and inverse variation. We have also provided some tips and tricks for working with joint and inverse variation. We hope that this article has been helpful in your understanding of joint and inverse variation.