Ladder Slip Eb51Bc

1. **State the problem:** A ladder of length $3$ m (which is $300$ cm) is resting against a vertical wall. Initially, the foot of the ladder is $70$ cm from the wall. The ladder slips, moving the foot $50$ cm further away, so the new distance from the wall is $70 + 50 = 120$ cm. We need to find how far down the wall the ladder slipped, i.e., the vertical distance the top of the ladder moved down. 2. **Set up the initial situation:** The ladder, wall, and ground form a right triangle with the ladder as the hypotenuse. Initial vertical height $h_1$ satisfies: $$ 300^2 = 70^2 + h_1^2 $$ 3. **Calculate initial height $h_1$:** $$ 300^2 = 90000 $$ $$ 70^2 = 4900 $$ $$ h_1^2 = 90000 - 4900 = 85100 $$ $$ h_1 = \sqrt{85100} \approx 291.7 \text{ cm} $$ 4. **Set up the situation after slipping:** The foot is now $120$ cm from the wall, ladder length remains $300$ cm. New height $h_2$ satisfies: $$ 300^2 = 120^2 + h_2^2 $$ Calculate $h_2$: $$ 120^2 = 14400 $$ $$ h_2^2 = 90000 - 14400 = 75600 $$ $$ h_2 = \sqrt{75600} \approx 274.9 \text{ cm} $$ 5. **Calculate how far the ladder slipped down the wall:** $$ \text{Slip distance} = h_1 - h_2 = 291.7 - 274.9 = 16.8 \text{ cm} $$ 6. **Final answer rounded to nearest cm:** $$ 17 \text{ cm} $$ **Answer:** The ladder slipped down the wall approximately $17$ cm.