A particle is moving along the curve . As the particle passes through the point , its -coordinate increases at a rate of units per second. Find the rate of change of the distance from the particle to the origin at this instant.
To find the rate of change of the distance from the particle to the origin at a particular instant, we can use the concept of the derivative.
Let's consider the particle's position at this instant as (x, y). Since we are only concerned with the distance from the particle to the origin, we can use the distance formula:
Distance = √(x^2 + y^2)
Now, let's differentiate both sides of the equation with respect to time (t) to find the rate of change of the distance:
d(Distance)/dt = d(√(x^2 + y^2))/dt
To evaluate this expression, we need to use the chain rule. Let's start by finding the partial derivatives:
∂(√(x^2 + y^2))/∂x = (1/2) * (x^2 + y^2)^(-1/2) * 2x
∂(√(x^2 + y^2))/∂y = (1/2) * (x^2 + y^2)^(-1/2) * 2y
Next, we'll substitute the given information into these derivatives:
∂(√(x^2 + y^2))/∂x = (1/2) * (x^2 + y^2)^(-1/2) * 2x = (1/2) * (x^2 + 4^2)^(-1/2) * 4 = (1/4) * (x^2 + 16)^(-1/2) * x
∂(√(x^2 + y^2))/∂y = (1/2) * (x^2 + y^2)^(-1/2) * 2y = (1/2) * (x^2 + 4^2)^(-1/2) * 0 = 0
Now, we can substitute these partial derivatives into the expression we derived earlier:
d(Distance)/dt = (1/4) * (x^2 + 16)^(-1/2) * x * dx/dt
Given that dx/dt is the rate at which the x-coordinate is changing, we can substitute this value:
d(Distance)/dt = (1/4) * (x^2 + 16)^(-1/2) * x * dx/dt = (1/4) * (16 + 16)^(-1/2) * 4 * dx/dt = dx/dt
Therefore, the rate of change of the distance from the particle to the origin at this instant is equal to the rate at which the x-coordinate is changing.