Answered

A small submarine starts at $-1,260$ feet in relation to sea level. Then it descends 5 feet per minute for 12 minutes. A student's work to determine the new position of the submarine in relation to sea level is shown.



Student's Work

$\begin{array}{l}

-1,260+(-5)(12)=-1,260+60 \\

=-1,200

\end{array}$



Which statement is true?


AThe student didn't add $-1,260$ and 60 correctly.
BThe student's work is correct.
CThe student didn't multiply -5 and 12 correctly.
DThe student didn't follow the order of operations.

Verified Answer :

  • The submarine's initial depth is \(-1,260\) feet, and its descent of \(5\) feet/minute for \(12\) minutes is a change of \(-5 \times 12 = -60\) feet.
  • The correct calculation for the final position is \(-1,260 - 60 = -1,320\) feet.
  • The student incorrectly calculated the product \((-5)(12)\) as \(+60\), when it should be \(-60\).
  • The true statement is: The student didn't multiply \(-5\) and \(12\) correctly.
  • Final Answer: \(\boxed{\text{The student didn't multiply -5 and 12 correctly.}}\)

Explanation

  1. Analyzing the Problem and Calculating Correct Position

The submarine's initial position is \(-1,260\) feet relative to sea level. It descends at a rate of \(5\) feet per minute for \(12\) minutes. Since the submarine is descending, this motion represents a decrease in depth, which corresponds to a change of \(-5 \text{ feet/minute} \times 12 \text{ minutes} = -60 \text{ feet}\). Therefore, the new position should be the initial position plus the change, which is \(-1,260 + (-60) = -1,320\) feet.

  1. Identifying the Student's Error

Looking at the student's work:
\(-1,260 + (-5)(12) = -1,260 + 60 = -1,200\).
The student correctly sets up the expression \(-1,260 + (-5)(12)\). However, in the next step, they simplify \((-5)(12)\) to \(+60\) instead of \(-60\). This incorrect sign results in an addition instead of a subtraction, leading to an incorrect final result of \(-1,200\).

  1. Evaluating the Options and Finalizing the Answer

Comparing our analysis to the provided options:

  • The student didn't add \(-1,260\) and \(60\) correctly: This is false, because the calculation \(-1,260 + 60 = -1,200\) is arithmetically correct, but the term \(60\) is wrong.
  • The student's work is correct: This is false.
  • The student didn't multiply \(-5\) and \(12\) correctly: This is true, as \((-5)(12)\) should be \(-60\), not \(60\).
  • The student didn't follow the order of operations: This is false, they performed the multiplication before the addition.

Therefore, the statement "The student didn't multiply \(-5\) and \(12\) correctly" is the true one.

Examples

Understanding how to handle signed numbers and rates of change is crucial in many real-world scenarios. For example, in accounting, if a company has an initial budget balance of \(\$1,260\) and it makes a recurring payment of \(\$5\) each month for \(12\) months, failing to correctly account for these as negative values (expenses) would lead to a significant miscalculation of the remaining balance. Correctly identifying that a 'descent' or 'payment' is a negative change allows for accurate financial and physical modeling.

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