Answered

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 & =k t^2 \\

12 & =k \cdot 4^2 \\

12 & =16 k \\

0.75 & =k \\

48 & =0.75 t^2 \\

64 & =t^2 \\

8 & =t

\end{aligned}$



What is the first error, if any, in Nick's work?


AHe did not make any errors.
BHe used an equation that models inverse variation instead of direct variation.
CHe substituted incorrectly when calculating the constant of variation.
DHe made an error when isolating $t^2$.

Verified Answer :

  • The relationship is defined as direct variation where \(s = kt^2\).
  • Using the given data \(s=4\) and \(t=12\), the correct substitution is \(4 = k(12)^2\).
  • Nick substituted incorrectly by using \(s=12\) and \(t=4\), which is the first error in his work.
  • The correct option is: He substituted incorrectly when calculating the constant of variation.

Explanation

  1. Analyzing the problem and identifying the first step

The problem states that the variable \(s\) varies directly as the square of \(t\). Mathematically, this relationship is expressed as \(s = kt^2\), where \(k\) is the constant of variation. We are given the initial conditions \(s = 4\) and \(t = 12\). To find the value of \(k\), we substitute these given values into the equation: \(4 = k(12)^2\). Analyzing Nick's work, we see he wrote the equation \(12 = k(4)^2\). This indicates that he swapped the given values of \(s\) and \(t\), substituting \(s = 12\) and \(t = 4\) instead of \(s = 4\) and \(t = 12\).

  1. Evaluating the error in Nick's work

Comparing Nick's first few lines of work to the correct setup:

  • Correct equation: \(s = kt^2\)
  • Correct substitution: \(4 = k(12)^2\)
  • Nick's equation: \(s = kt^2\)
  • Nick's substitution: \(12 = k(4)^2\)

By comparing these, it is clear that Nick's first error occurred during the substitution process used to determine the constant of variation \(k\). His subsequent steps (finding \(k = 0.75\) and solving for \(t\) given \(s=48\)) were calculated correctly based on his incorrect initial substitution, but the process became flawed from the moment the values were incorrectly placed into the formula.

  1. Conclusion and summary of the error

Since the initial substitution of \(s=12\) and \(t=4\) is the first departure from the correct mathematical procedure dictated by the problem statement, we conclude that Nick's first error was substituting incorrectly when calculating the constant of variation. The correct steps would have been:

  1. \(s = kt^2\)
  2. \(4 = k(12)^2 \Rightarrow 4 = 144k \Rightarrow k = \frac{4}{144} = \frac{1}{36}\)
  3. Using \(k = \frac{1}{36}\), we find \(t\) when \(s=48\): \(48 = \frac{1}{36}t^2 \Rightarrow 1728 = t^2 \Rightarrow t = \sqrt{1728} \approx 41.57\).

Therefore, the first error is that he substituted incorrectly.

Examples

In physics and engineering, identifying the correct input variables for a formula is crucial, especially when dealing with physical constants. If you are calculating the drag force on an object which is proportional to the square of its velocity (\(F_d = kv^2\)), misidentifying the measured force and velocity values will lead to an incorrect drag coefficient. This error would cause inaccurate predictions for all future velocity or force calculations using that specific coefficient, illustrating why verifying data input is as vital as the calculation itself.

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