A force `2hati+3hatj+4hatk` N acts on a body for 4 sec, produces a displacement of `(3hati+4hatj+5hatk)`m. the power used is

Options

A. 9.5 W
B. 7.5 W
C. 6.5 W
D. 4.5 W

Explanation

To find the power used when a force acts on a body and produces a displacement, we can follow these steps:

### Step 1: Identify the given quantities
We have:
- Force \( \vec{F} = 2\hat{i} + 3\hat{j} + 4\hat{k} \) N
- Displacement \( \vec{S} = 3\hat{i} + 4\hat{j} + 5\hat{k} \) m
- Time \( t = 4 \) seconds

### Step 2: Calculate the work done
The work done \( W \) by a force is given by the dot product of the force vector and the displacement vector:
\[ W = \vec{F} \cdot \vec{S} \]
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 formula for the dot product:
\[ W = (2 \times 3) + (3 \times 4) + (4 \times 5) \]
Calculating each term:
\[ W = 6 + 12 + 20 = 38 \text{ Joules} \]

### Step 3: Calculate the power
Power \( P \) is defined as the work done per unit time:
\[ P = \frac{W}{t} \]
Substituting the values we found:
\[ P = \frac{38 \text{ Joules}}{4 \text{ seconds}} = 9.5 \text{ Watts} \]

### Final Answer
The power used is \( 9.5 \) Watts.

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