A particle is moving along the curve y= 2 \sqrt{4 x + 4}. As the particle passes through the point (3, 8), its x-coordinate increases at a rate of 5 units per second. Find the rate of change of the distance from the particle to the origin at this instant.

Question

A particle is moving along the curve y= 2 \sqrt{4 x + 4}. As the particle passes through the point (3, 8), its x-coordinate increases at a rate of 5 units per second. Find the rate of change of the distance from the particle to the origin at this instant.

Answer 1

To find the rate of change of the distance from the particle to the origin, we need to find the distance function between the particle and the origin, and then differentiate it with respect to time.

Let's start by finding the distance function. The distance from the particle to the origin can be calculated using the distance formula:

distance = √((x - 0)^2 + (y - 0)^2)
distance = √(x^2 + y^2)

Now, we need to express the distance in terms of x only. Since we are given the equation of the curve y = 2 \sqrt(4x + 4), we can substitute this expression into the distance formula:

distance = √(x^2 + (2 \sqrt(4x + 4))^2)
distance = √(x^2 + 4(4x + 4))
distance = √(x^2 + 16x + 16)

Now, we have the expression for the distance in terms of x. To find the rate of change of the distance from the particle to the origin, we need to differentiate this expression with respect to time, taking into account that the x-coordinate of the particle is increasing at a rate of 5 units per second.

Let's differentiate the distance expression with respect to time:

d(distance)/dt = d/dt √(x^2 + 16x + 16)

To differentiate this, we can use the chain rule. Let's denote dx/dt as the rate of change of the x-coordinate, which is given as 5 units per second:

d(distance)/dt = (1/2) * (x^2 + 16x + 16)^(-1/2) * d/dt(x^2 + 16x + 16)

Now, substitute dx/dt with 5:

d(distance)/dt = (1/2) * √(x^2 + 16x + 16) * (d/dt(x^2 + 16x + 16))
d(distance)/dt = (1/2) * √(x^2 + 16x + 16) * (2x + 16) * (dx/dt)
d(distance)/dt = √(x^2 + 16x + 16) * (x + 8) * (dx/dt)

Finally, substitute the given x-coordinate and the rate of change of the x-coordinate:

d(distance)/dt = √((3)^2 + 16(3) + 16) * (3 + 8) * 5

Simplifying this expression will give you the rate of change of the distance from the particle to the origin at the instant when the particle passes through the point (3, 8).