Select The Correct Answer.The Variable $s$ Varies Directly As The Square Of $t$. When $s=4$, $t=12$. Nick's Work Finding The Value Of $t$ When $s=48$ Is Shown:$[ \begin{aligned} s &

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Introduction

In mathematics, direct variation is a fundamental concept that describes the relationship between two variables. It states that as one variable increases or decreases, the other variable also increases or decreases in a consistent manner. In this article, we will explore the concept of direct variation, specifically focusing on the relationship between the variables $s$ and $t$, where $s$ varies directly as the square of $t$. We will also analyze Nick's work in finding the value of $t$ when $s=48$.

Understanding Direct Variation

Direct variation is a type of linear relationship between two variables. It can be represented mathematically as $y = kx$, where $y$ is the dependent variable, $x$ is the independent variable, and $k$ is the constant of variation. In the case of direct variation, the relationship between the variables is proportional, meaning that as one variable increases or decreases, the other variable also increases or decreases in a consistent manner.

The Relationship Between $s$ and $t$

In this problem, we are given that $s$ varies directly as the square of $t$. This can be represented mathematically as $s = kt^2$, where $k$ is the constant of variation. We are also given that when $s=4$, $t=12$. Using this information, we can find the value of $k$ by substituting the values of $s$ and $t$ into the equation.

Finding the Value of $k$

Substituting $s=4$ and $t=12$ into the equation $s = kt^2$, we get:

4=k(12)24 = k(12)^2

Simplifying the equation, we get:

4=k(144)4 = k(144)

Dividing both sides of the equation by 144, we get:

k=4144k = \frac{4}{144}

Simplifying the fraction, we get:

k=136k = \frac{1}{36}

Nick's Work in Finding the Value of $t$

Nick's work in finding the value of $t$ when $s=48$ is shown below:

s=kt248=136t248×36=t21728=t21728=t1728=41.569\begin{aligned} s &= kt^2 \\ 48 &= \frac{1}{36}t^2 \\ 48 \times 36 &= t^2 \\ 1728 &= t^2 \\ \sqrt{1728} &= t \\ \sqrt{1728} &= 41.569 \end{aligned}

However, Nick's work contains an error. The correct value of $t$ can be found by taking the square root of both sides of the equation $t^2 = 1728$.

Correcting Nick's Work

To correct Nick's work, we need to take the square root of both sides of the equation $t^2 = 1728$. This gives us:

t=1728t = \sqrt{1728}

Simplifying the square root, we get:

t=41.569t = 41.569

However, we can simplify the square root further by finding the prime factorization of 1728.

Prime Factorization of 1728

The prime factorization of 1728 is:

1728=26×331728 = 2^6 \times 3^3

Using the prime factorization, we can simplify the square root as follows:

t=26×33t = \sqrt{2^6 \times 3^3}

Simplifying the square root, we get:

t=23×3×2t = 2^3 \times 3 \times \sqrt{2}

Simplifying further, we get:

t=242t = 24\sqrt{2}

Conclusion

In conclusion, we have explored the concept of direct variation and its applications in mathematics. We have analyzed the relationship between the variables $s$ and $t$, where $s$ varies directly as the square of $t$. We have also corrected Nick's work in finding the value of $t$ when $s=48$. The correct value of $t$ is $24\sqrt{2}$.

Final Answer

Q: What is direct variation?

A: Direct variation is a type of linear relationship between two variables. It can be represented mathematically as $y = kx$, where $y$ is the dependent variable, $x$ is the independent variable, and $k$ is the constant of variation.

Q: How is direct variation different from other types of variation?

A: Direct variation is different from other types of variation, such as inverse variation and joint variation, in that the relationship between the variables is proportional. In direct variation, as one variable increases or decreases, the other variable also increases or decreases in a consistent manner.

Q: What is the constant of variation in direct variation?

A: The constant of variation in direct variation is a number that represents the rate at which the dependent variable changes in response to changes in the independent variable. It is denoted by the symbol $k$ and is a key component of the direct variation equation.

Q: How do you find the constant of variation in direct variation?

A: To find the constant of variation in direct variation, you need to know the values of the dependent and independent variables. You can then use the direct variation equation to solve for the constant of variation.

Q: What is the relationship between the variables in direct variation?

A: In direct variation, the relationship between the variables is proportional. This means that as one variable increases or decreases, the other variable also increases or decreases in a consistent manner.

Q: How do you determine if a relationship is direct variation?

A: To determine if a relationship is direct variation, you need to check if the relationship is proportional. You can do this by graphing the relationship and checking if the graph is a straight line. You can also use the direct variation equation to check if the relationship is proportional.

Q: What are some real-world applications of direct variation?

A: Direct variation has many real-world applications, including:

  • Physics: Direct variation is used to describe the relationship between the distance traveled by an object and the time it takes to travel that distance.
  • Engineering: Direct variation is used to describe the relationship between the force applied to an object and the distance it moves.
  • Economics: Direct variation is used to describe the relationship between the price of a good and the quantity demanded.

Q: How do you solve direct variation problems?

A: To solve direct variation problems, you need to use the direct variation equation and the given values of the dependent and independent variables. You can then solve for the constant of variation and use it to find the value of the dependent variable.

Q: What are some common mistakes to avoid when solving direct variation problems?

A: Some common mistakes to avoid when solving direct variation problems include:

  • Not using the correct equation
  • Not substituting the given values into the equation
  • Not solving for the constant of variation
  • Not using the correct units

Q: How do you check your work when solving direct variation problems?

A: To check your work when solving direct variation problems, you need to:

  • Graph the relationship and check if it is a straight line
  • Use the direct variation equation to check if the relationship is proportional
  • Check if the units are correct
  • Check if the solution is reasonable

Conclusion

In conclusion, direct variation is a fundamental concept in mathematics that describes the relationship between two variables. It has many real-world applications and is used to solve a wide range of problems. By understanding direct variation and how to solve direct variation problems, you can develop a deeper understanding of the world around you.