A body, under the action of a force `F=6hat(i)-8hat(j)+10hat(k)` acceleration of `1ms^(-2)`. The mass of this body must be

Options

A. `2sqrt(10)` kg
B. 10 kg
C. 20 kg
D. `10sqrt(2)` kg

Answer

To find the mass of the body under the action of the force \( \mathbf{F} = 6\hat{i} - 8\hat{j} + 10\hat{k} \) and given an acceleration of \( 1 \, \text{m/s}^2 \), we can use Newton's second law of motion, which states:

\[ \mathbf{F} = m \cdot \mathbf{a} \]

Where:
- \( \mathbf{F} \) is the force vector,
- \( m \) is the mass of the body,
- \( \mathbf{a} \) is the acceleration vector.

### Step 1: Calculate the Magnitude of the Force
First, we need to calculate the magnitude of the force vector \( \mathbf{F} \):

\[ |\mathbf{F}| = \sqrt{(6)^2 + (-8)^2 + (10)^2} \]

Calculating each term:

\[ |\mathbf{F}| = \sqrt{36 + 64 + 100} \]

Now, sum these values:

\[ |\mathbf{F}| = \sqrt{200} \]

### Step 2: Simplify the Magnitude
We can simplify \( \sqrt{200} \):

\[ |\mathbf{F}| = \sqrt{100 \cdot 2} = 10\sqrt{2} \]

### Step 3: Use Newton's Second Law
According to Newton's second law, we can express the force in terms of mass and acceleration:

\[ |\mathbf{F}| = m \cdot |\mathbf{a}| \]

Given that the acceleration \( |\mathbf{a}| = 1 \, \text{m/s}^2 \):

\[ 10\sqrt{2} = m \cdot 1 \]

### Step 4: Solve for Mass
Now, we can solve for the mass \( m \):

\[ m = 10\sqrt{2} \, \text{kg} \]

### Final Answer
Thus, the mass of the body is:

\[ m = 10\sqrt{2} \, \text{kg} \]

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