Question

select the correct answer.

the variable \\(s\\) varies directly as the square of \\(t\\). when \\(s = 4\\), \\(t = 12\\). nicks work finding the value of \\(t\\) when \\(s = 48\\) is shown:

\\
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$$\begin{aligned} s &= kt^2 \\\\ 12 &= k \\cdot 4^2 \\\\ 12 &= 16k \\\\ 0.75 &= k \\\\ 48 &= 0.75t^2 \\\\ 64 &= t^2 \\\\ 8 &= t \\end{aligned}$$

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what is the first error, if any, in nicks work?

• he did not make any errors.
• he used an equation that models inverse variation instead of direct variation.
• he substituted incorrectly when calculating the constant of variation.
• he made an error when isolating \\(t^2\\).

Explanation:

Analyze the given variation relationship

$$ s = k t^2 $$

Evaluate Nick's substitution step

Given values: \(s = 4\) and \(t = 12\).
Nick's substitution:

$$ 12 = k \cdot 4^2 $$

This is incorrect because he substituted \(s = 12\) and \(t = 4\), which swaps the values of \(s\) and \(t\).

Identify the corresponding error option

The first error is substituting incorrectly when calculating the constant of variation \(k\).

Answer:

• He did not make any errors.
• He used an equation that models inverse variation instead of direct variation.
• He substituted incorrectly when calculating the constant of variation. (Correct answer)
• He made an error when isolating \(t^2\).