Question

A 25 ft long ladder is placed against the wall with its base 7 ft from the wall. The base of the ladder is drawn out so that the top comes down by half the distance that the base is drawn out. This distance is in the range:

• A. (2, 7)
• B. (5, 8)
• C. (9, 10)
• D. (3, 7)
• E. None of the above

Hint:

In ladder problems, carefully account for the relationship between base extension and top descent, then apply Pythagoras for the new triangle.

Answer 1

Answer: (E)

Step 1: Initial setup.
- Ladder length = 25 ft. - Base initially from wall = 7 ft. - Height of top initially: \[ \sqrt{25^2 - 7^2} = \sqrt{625 - 49} = \sqrt{576} = 24 \; \text{ft} \]

Step 2: New position.
Let the base be drawn outward by \(x\). Then: - New base distance = \(7 + x\). - Top descends by \(\frac{x}{2}\). - New height = \(24 - \frac{x}{2}\).

Step 3: Apply Pythagoras.
\[ (7+x)^2 + \left(24 - \frac{x}{2}\right)^2 = 25^2 \] Expanding: \[ (49 + 14x + x^2) + (576 - 24x + \frac{x^2}{4}) = 625 \] \[ 625 + (-10x) + \frac{5}{4}x^2 = 625 \] \[ \frac{5}{4}x^2 - 10x = 0 \] \[ \Rightarrow x \left(\frac{5}{4}x - 10\right) = 0 \] \[ \Rightarrow x = 0 \quad \text{or} \quad x = 8 \]

Step 4: Interpret result.
- \(x=0\) is trivial (no movement). - \(x=8\) ft is the valid solution. So, the distance drawn out is **exactly 8 ft**.

Step 5: Compare with given options.
The value 8 does not fall into any of the provided ranges.

Final Answer:
\[ \boxed{\text{None of the above}} \]