The coordinates of a moving particle at any time't' are given by `x = alphat^(3)` and `y = betat^(3)`. The speed of the particle at time 't' is given by

Options

A. `t^(2)sqrt(alpha^(2)+beta^(2))`
B. `sqrt(alpha^(2)+beta^(2))`
C. `3tsqrt(alpha^(2)+beta^(2))`
D. `3t^(2)sqrt(alpha^(2)+beta^(2))`

Explanation

To find the speed of the particle at time 't', we will follow these steps:

### Step 1: Differentiate the position equations
The coordinates of the particle are given by:
- \( x(t) = \alpha t^3 \)
- \( y(t) = \beta t^3 \)

We need to find the velocity components \( v_x \) and \( v_y \) by differentiating these equations with respect to time \( t \).

### Step 2: Calculate \( v_x \)
To find \( v_x \):
\[ v_x = \frac{dx}{dt} = \frac{d}{dt}(\alpha t^3) = 3\alpha t^2 \]

### Step 3: Calculate \( v_y \)
To find \( v_y \):
\[ v_y = \frac{dy}{dt} = \frac{d}{dt}(\beta t^3) = 3\beta t^2 \]

### Step 4: Calculate the speed \( v \)
The speed of the particle is given by the formula:
\[ v = \sqrt{v_x^2 + v_y^2} \]
Substituting the expressions for \( v_x \) and \( v_y \):
\[ v = \sqrt{(3\alpha t^2)^2 + (3\beta t^2)^2} \]

### Step 5: Simplify the expression
Now, simplify the expression inside the square root:
\[ v = \sqrt{9\alpha^2 t^4 + 9\beta^2 t^4} = \sqrt{9(\alpha^2 + \beta^2)t^4} \]
\[ v = 3t^2\sqrt{\alpha^2 + \beta^2} \]

### Final Result
Thus, the speed of the particle at time \( t \) is:
\[ v = 3t^2\sqrt{\alpha^2 + \beta^2} \]

---