A ladder 5 m in length is resting against vertical wall. The bottom of the ladder is pulled along the ground away from the wall at the rate of 1.5m/sec. Then the highest point of the ladder when the foot of the ladder is 4.0 m away from the wall , decreases at the rate of

Options

A. 2m/sec
B. 3m/sec
C. 2.5m/sec
D. 1.5m/sec

Explanation

To solve the problem, we will use the Pythagorean theorem and implicit differentiation. Let's break it down step by step:

### Step 1: Understand the Problem
We have a ladder of length 5 m resting against a vertical wall. The bottom of the ladder is being pulled away from the wall at a rate of 1.5 m/s. We need to find how fast the top of the ladder is descending when the bottom of the ladder is 4 m away from the wall.

### Step 2: Set Up the Variables
Let:
- \( x \) = distance from the wall to the bottom of the ladder
- \( y \) = height of the top of the ladder above the ground
- The length of the ladder is constant at 5 m.

From the Pythagorean theorem, we know:
\[ x^2 + y^2 = 5^2 \]
or
\[ x^2 + y^2 = 25 \]

### Step 3: Differentiate with Respect to Time
We differentiate both sides of the equation with respect to time \( t \):
\[ \frac{d}{dt}(x^2 + y^2) = \frac{d}{dt}(25) \]
This gives us:
\[ 2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0 \]
We can simplify this to:
\[ x \frac{dx}{dt} + y \frac{dy}{dt} = 0 \]

### Step 4: Substitute Known Values
We know:
- \( \frac{dx}{dt} = 1.5 \, \text{m/s} \) (the rate at which the bottom of the ladder is moving away from the wall)
- When \( x = 4 \, \text{m} \), we need to find \( y \).

Using the Pythagorean theorem:
\[ 4^2 + y^2 = 25 \implies 16 + y^2 = 25 \implies y^2 = 9 \implies y = 3 \, \text{m} \]

Now, substitute \( x = 4 \), \( y = 3 \), and \( \frac{dx}{dt} = 1.5 \) into the differentiated equation:
\[ 4(1.5) + 3 \frac{dy}{dt} = 0 \]

### Step 5: Solve for \( \frac{dy}{dt} \)
This simplifies to:
\[ 6 + 3 \frac{dy}{dt} = 0 \]
\[ 3 \frac{dy}{dt} = -6 \]
\[ \frac{dy}{dt} = -2 \, \text{m/s} \]

### Conclusion
The highest point of the ladder is descending at a rate of 2 m/s when the foot of the ladder is 4 m away from the wall.