A particle moves along the straight line `y= 3x+5`. Which coordinate changes at a faster rate ?

Options

A. x- coordinate
B. y- coordinate
C. Both x and y coordinates
D. Data insufficient

Explanation

To determine which coordinate changes at a faster rate for a particle moving along the straight line given by the equation \( y = 3x + 5 \), we can follow these steps:

### Step 1: Understand the relationship between x and y
The equation \( y = 3x + 5 \) shows that y is a linear function of x. The slope of the line is 3, which indicates how y changes with respect to x.

### Step 2: Differentiate the equation with respect to time
To find how the coordinates change with respect to time, we differentiate both sides of the equation with respect to time \( t \):

\[ \frac{dy}{dt} = \frac{d}{dt}(3x + 5) \]

### Step 3: Apply the chain rule
Using the chain rule, we differentiate the right-hand side:

\[ \frac{dy}{dt} = 3 \frac{dx}{dt} + 0 \]

The derivative of the constant (5) is zero.

### Step 4: Simplify the equation
This simplifies to:

\[ \frac{dy}{dt} = 3 \frac{dx}{dt} \]

### Step 5: Compare the rates of change
From the equation \( \frac{dy}{dt} = 3 \frac{dx}{dt} \), we can see that the rate of change of y with respect to time is three times the rate of change of x with respect to time. This indicates that y changes at a faster rate than x.

### Conclusion
Thus, the y-coordinate changes at a faster rate than the x-coordinate.

### Final Answer
The y-coordinate changes at a faster rate.

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