Question 1174226
Absolutely! Let's solve this problem step-by-step.

**1. Set up the Problem**

* **Ladder Length:** 4 meters (constant)
* **Distance from Wall (D):** 2.5 meters
* **Height on Wall (H):** We need to find this.
* **Rate of Change:** We want to find dH/dD (how the height changes with respect to the distance).

**2. Use the Pythagorean Theorem**

* We have a right triangle formed by the wall, the ground, and the ladder.
* The Pythagorean theorem states: D² + H² = 4² (where 4 is the ladder length)
* D² + H² = 16

**3. Express Height (H) in Terms of Distance (D)**

* H² = 16 - D²
* H = √(16 - D²)

**4. Find the Instantaneous Rate of Change (dH/dD)**

* We'll use the given central difference method to estimate the derivative.
* The formula for the central difference is:
    * dH/dD ≈ [H(D + h) - H(D - h)] / (2h)
* Where:
    * D = 2.5 meters
    * h = 0.01 meters

**5. Calculate H(D + h) and H(D - h)**

* H(D + h) = H(2.5 + 0.01) = H(2.51) = √(16 - 2.51²) ≈ √(16 - 6.3001) ≈ √9.6999 ≈ 3.11446
* H(D - h) = H(2.5 - 0.01) = H(2.49) = √(16 - 2.49²) ≈ √(16 - 6.2001) ≈ √9.7999 ≈ 3.13048

**6. Substitute into the Central Difference Formula**

* dH/dD ≈ (3.11446 - 3.13048) / (2 * 0.01)
* dH/dD ≈ (-0.01602) / 0.02
* dH/dD ≈ -0.801

**Result**

* The estimated instantaneous rate of change of the height of the ladder with respect to the distance of the bottom of the ladder from the wall is approximately -0.801 meters per meter.

**Therefore, when the bottom of the ladder is 2.5 meters from the wall, the top of the ladder is sliding down the wall at a rate of approximately 0.801 meters for every meter the bottom slides away.**