Question

The market value of beams, made of a rare metal, has a unique property: the market value of any such beam is proportional to the square of its length. Due to an accident, one such beam got broken into two pieces having lengths in the ratio 4:9. Considering each broken piece as a separate beam, how much gain or loss, with respect to the market value of the original beam before the accident, is incurred?

• A. 74.23% gain
• B. No gain or loss
• C. 31.77% loss
• D. 42.60% loss
• E. 57.40% loss

Hint:

When dealing with proportional changes in values, square the lengths if the value is proportional to the square of the length.

Answer 1

The Correct Option is A

Step 1: Establish the original beam's length and value.
Let the original beam's length be \( L \). Its market value is proportional to \( L^2 \), represented as \( kL^2 \), where \( k \) is the constant of proportionality.The beam is divided into two segments with lengths \( 4x \) and \( 9x \). The market value of the first segment is proportional to \( (4x)^2 = 16x^2 \), and the second segment's value is proportional to \( (9x)^2 = 81x^2 \).
Step 2: Determine the aggregate value of the fragmented pieces.
The combined market value of the two pieces is:\[16x^2 + 81x^2 = 97x^2\]
Step 3: Compare the initial and final values.
The initial value was \( kL^2 \). Given that \( L = 13x \) (as the sum of the lengths of the broken pieces equals the original length), the initial value is:\[kL^2 = k(13x)^2 = 169k x^2\]The total market value post-fracture is \( 97k x^2 \). The difference in value is:\[\text{Difference} = 169k x^2 - 97k x^2 = 72k x^2\]
Step 4: Quantify the percentage change.
The percentage gain is calculated as:\[\frac{72k x^2}{97k x^2} \times 100 = 74.23% \text{ gain}\]
Final Answer: \[\boxed{74.23% \text{ gain}}\]