A force of `(2hat(i) + 3hat(j) + 4hat(k))N` acts on a body for 19s and produces a displacement of `(3 hat(i) + 4hat(j) + 5hat(k))m`. The power consumed as

Options

A. 2W
B. 6W
C. 19W
D. 38W

Explanation

To find the power consumed by the force acting on the body, we can follow these steps:

### Step 1: Identify the Force and Displacement
The force acting on the body is given as:
\[ \mathbf{F} = 2\hat{i} + 3\hat{j} + 4\hat{k} \text{ N} \]
The displacement of the body is given as:
\[ \mathbf{S} = 3\hat{i} + 4\hat{j} + 5\hat{k} \text{ m} \]

### Step 2: Calculate the Work Done
The work done \( W \) by the force is calculated using the dot product of the force and displacement vectors:
\[ W = \mathbf{F} \cdot \mathbf{S} \]
Calculating the dot product:
\[ W = (2\hat{i} + 3\hat{j} + 4\hat{k}) \cdot (3\hat{i} + 4\hat{j} + 5\hat{k}) \]
Using the properties of dot products:
\[ W = (2 \cdot 3) + (3 \cdot 4) + (4 \cdot 5) \]
Calculating each term:
\[ W = 6 + 12 + 20 = 38 \text{ J} \]

### Step 3: Calculate the Power
Power \( P \) is defined as the work done per unit time. Given that the force acts for \( t = 19 \) seconds, we can calculate the power:
\[ P = \frac{W}{t} = \frac{38 \text{ J}}{19 \text{ s}} = 2 \text{ W} \]

### Final Answer
The power consumed is:
\[ \boxed{2 \text{ W}} \]

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