Question

If the cube of n is 180 greater than the square of n, then n =

• A. 10
• B. 9
• C. 8
• D. 7
• E. 6

Hint:

When a question asks you to solve an equation and provides multiple-choice integer answers, back-solving (plugging the answer choices into the equation) is often the fastest and most reliable strategy, especially for higher-order polynomials like cubics.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This question requires translating a word problem into an algebraic equation and then solving for the variable \(n\).
Step 2: Key Formula or Approach:
The problem statement can be written as an equation:
\[ n^3 = n^2 + 180 \] This can be rearranged into a cubic equation: \(n^3 - n^2 - 180 = 0\). Since solving a cubic equation algebraically is complex, the most efficient method for a multiple-choice question is to test the given integer options (back-solving).
Step 3: Detailed Explanation:
We will substitute each option for \(n\) into the expression \(n^3 - n^2\) and check if the result is 180.
- (A) If \(n=10\), then \(10^3 - 10^2 = 1000 - 100 = 900\). This is not 180.
- (B) If \(n=9\), then \(9^3 - 9^2 = 729 - 81 = 648\). This is not 180.
- (C) If \(n=8\), then \(8^3 - 8^2 = 512 - 64 = 448\). This is not 180.
- (D) If \(n=7\), then \(7^3 - 7^2 = 343 - 49 = 294\). This is not 180.
- (E) If \(n=6\), then \(6^3 - 6^2 = 216 - 36 = 180\). This matches the condition.
Step 4: Final Answer:
The value of \(n\) that satisfies the condition is 6.