A particle moves in a plane with a constant speed along a path `y=2x^(2)+3x-4` When the particle is at (0,-4) the direction along which it is moving is inclined to the X axis at an angle. Given `tan^(-1) 3=72^(@)`

Options

A. `63^(@)`
B. `72^(@)`
C. `27^(@)`
D. `0^(@)`

Explanation

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the given path
The path of the particle is given by the equation:
\[ y = 2x^2 + 3x - 4 \]

### Step 2: Find the derivative of the path
To find the direction of the particle at a given point, we need to calculate the derivative \( \frac{dy}{dx} \). This derivative represents the slope of the tangent to the curve at any point \( (x, y) \).

Differentiating the equation with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(2x^2 + 3x - 4) = 4x + 3 \]

### Step 3: Evaluate the derivative at the point (0, -4)
Now, we need to evaluate the derivative at the specific point where the particle is located, which is \( (0, -4) \).

Substituting \( x = 0 \) into the derivative:
\[ \frac{dy}{dx} \bigg|_{(0, -4)} = 4(0) + 3 = 3 \]

### Step 4: Relate the derivative to the angle
The slope of the tangent line at the point gives us the tangent of the angle \( \theta \) that the direction of the particle makes with the x-axis. Therefore:
\[ \tan \theta = 3 \]

### Step 5: Find the angle \( \theta \)
To find the angle \( \theta \), we take the arctangent (inverse tangent) of 3:
\[ \theta = \tan^{-1}(3) \]

### Step 6: Use the given information
It is given in the problem that:
\[ \tan^{-1}(3) = 72^\circ \]
Thus, we conclude that:
\[ \theta = 72^\circ \]

### Final Answer
The direction along which the particle is moving is inclined to the x-axis at an angle of \( 72^\circ \).

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