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

Understanding Direct Variation

Direct variation is a relationship between two variables where one variable is a constant multiple of the other. In mathematical terms, if $y$ varies directly as $x$, then $y = kx$ for some constant $k$. In this problem, we are given that the variable $s$ varies directly as the square of $t$, which can be written as $s = kt^2$.

Given Information

We are given that when $s = 4$, $t = 12$. This information can be used to find the value of the constant $k$.

Finding the Value of $k$

To find the value of $k$, we can substitute the given values of $s$ and $t$ into the equation $s = kt^2$. This gives us:

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

Simplifying the equation, we get:

$$4 = k(144)$$

Dividing both sides by 144, we get:

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

Simplifying further, we get:

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

Nick's Work

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

$$\begin{aligned} s & = kt^2 \\ 48 & = \frac{1}{36}t^2 \\ 48 \times 36 & = t^2 \\ 1728 & = t^2 \\ \sqrt{1728} & = t \\ \sqrt{144 \times 12} & = t \\ 12\sqrt{12} & = t \\ 12\sqrt{4 \times 3} & = t \\ 12 \times 2\sqrt{3} & = t \\ 24\sqrt{3} & = t \end{aligned}$$

Evaluating Nick's Work

Nick's work is correct, but it can be simplified further. We can start by simplifying the square root of 1728:

$$\sqrt{1728} = \sqrt{144 \times 12}$$

This can be further simplified by recognizing that 144 is a perfect square:

$$\sqrt{144 \times 12} = \sqrt{144} \times \sqrt{12}$$

Simplifying the square root of 144, we get:

$$\sqrt{144} = 12$$

So, we can rewrite the equation as:

$$12\sqrt{12} = t$$

We can further simplify the square root of 12 by recognizing that it is a perfect square:

$$\sqrt{12} = \sqrt{4 \times 3}$$

Simplifying the square root of 4, we get:

$$\sqrt{4} = 2$$

So, we can rewrite the equation as:

$$12 \times 2\sqrt{3} = t$$

Simplifying further, we get:

$$24\sqrt{3} = t$$

Conclusion

$$$$$$

Q: What is direct variation?

A: Direct variation is a relationship between two variables where one variable is a constant multiple of the other. In mathematical terms, if $y$ varies directly as $x$, then $y = kx$ for some constant $k$.

Q: What is the equation for direct variation when one variable varies as the square of another?

A: The equation for direct variation when one variable varies as the square of another is $y = kx^2$.

Q: How do we find the value of the constant $k$ in a direct variation problem?

A: To find the value of the constant $k$ in a direct variation problem, we can use the given information to substitute into the equation and solve for $k$.

Q: What is the relationship between the variables $s$ and $t$ in the problem?

A: The variable $s$ varies directly as the square of the variable $t$, which can be written as $s = kt^2$.

Q: How do we find the value of $t$ when $s = 48$?

A: To find the value of $t$ when $s = 48$, we can substitute the value of $s$ into the equation $s = kt^2$ and solve for $t$.

Q: What is the value of $t$ when $s = 48$?

A: The value of $t$ when $s = 48$ is $24\sqrt{3}$.

Q: How do we simplify the square root of 1728?

A: We can simplify the square root of 1728 by recognizing that 144 is a perfect square and that 12 is a perfect square.

Q: What is the simplified form of the square root of 1728?

A: The simplified form of the square root of 1728 is $12\sqrt{12}$.

Q: How do we further simplify the square root of 12?

A: We can further simplify the square root of 12 by recognizing that it is a perfect square.

Q: What is the simplified form of the square root of 12?

A: The simplified form of the square root of 12 is $2\sqrt{3}$.

Q: What is the final simplified form of the value of $t$ when $s = 48$?

A: The final simplified form of the value of $t$ when $s = 48$ is $24\sqrt{3}$.

Q: What is the relationship between the variables $s$ and $t$ in the problem?

A: The variable $s$ varies directly as the square of the variable $t$, which can be written as $s = kt^2$.

Q: How do we find the value of the constant $k$ in a direct variation problem?

A: To find the value of the constant $k$ in a direct variation problem, we can use the given information to substitute into the equation and solve for $k$.

Q: What is the equation for direct variation when one variable varies as the square of another?

A: The equation for direct variation when one variable varies as the square of another is $y = kx^2$.

Q: What is direct variation?

A: Direct variation is a relationship between two variables where one variable is a constant multiple of the other. In mathematical terms, if $y$ varies directly as $x$, then $y = kx$ for some constant $k$.