Find the number whose square equals the difference between 75.15^2 and 60.12^2 (i.e., (75.15)^2 − (60.12)^2).

Introduction:This problem uses the identity a^2 − b^2 = (a − b)(a + b). We then take the square root of the result to get the required number.

Given Data / Assumptions:

• a = 75.15
• b = 60.12
• We seek N such that N^2 = a^2 − b^2

Concept / Approach:Compute a − b and a + b, multiply, and then take the positive square root (principal value) since the context refers to a magnitude.

Step-by-Step Solution:a − b = 75.15 − 60.12 = 15.03a + b = 75.15 + 60.12 = 135.27a^2 − b^2 = (a − b)(a + b) = 15.03 * 135.27 = 2033.1081N = sqrt(2033.1081) = 45.09

Verification / Alternative check:Squaring 45.09 gives 2033.1081, matching a^2 − b^2 exactly.

Why Other Options Are Wrong:225.9: This is (a + b) * (a − b) divided or scaled incorrectly.67.635: Likely half of 135.27; not the square root of the product.15.03: Equals a − b, not sqrt((a − b)(a + b)).

Common Pitfalls:Mistaking a − b for the final answer; ignoring the square root step after using the identity.

Final Answer:45.09