A particle is moving along the curve y= 5sqrt{4x + 5}. As the particle passes through the point (1, 15), its x-coordinate increases at a rate of 2 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, we can use the Pythagorean theorem. Let's call the distance from the particle to the origin "r".
From the equation y = 5sqrt(4x + 5), we can solve for x in terms of y:
y = 5sqrt(4x + 5)
y/5 = sqrt(4x + 5)
(y/5)^2 = 4x + 5
(y^2)/25 - 5/25 = 4x
(y^2 - 5)/25 = 4x
x = (y^2 - 5)/100
Now, let's find the rate of change of x with respect to y using implicit differentiation:
dx/dy = (1/100) * 2y
dx/dy = y/50
Since the x-coordinate is increasing at a rate of 2 units per second, we have:
dx/dt = 2
To find dy/dt (the rate of change of y with respect to time), we can take the derivative of y = 5sqrt(4x + 5) with respect to t:
dy/dt = (dy/dx) * (dx/dt)
dy/dt = (dy/dx) * (dx/dt)
dy/dt = (dy/dx) * 2
Using dx/dy = y/50, we can solve for dy/dt in terms of y:
dy/dt = (y/50) * 2
dy/dt = y/25
Finally, we can find dr/dt (the rate of change of the distance from the particle to the origin) using the Pythagorean theorem:
r^2 = x^2 + y^2
Differentiating both sides with respect to t, we get:
2r * (dr/dt) = 2x * (dx/dt) + 2y * (dy/dt)
Since we are interested in finding dr/dt when the particle is at the point (1, 15), we can substitute x = 1 and y = 15 into the equation:
2r * (dr/dt) = 2(1) * (2) + 2(15) * (15/25)
Simplifying the equation, we have:
2r * (dr/dt) = 4 + 2(3)(3)
2r * (dr/dt) = 4 + 18
2r * (dr/dt) = 22
Since we want to find dr/dt, we can solve for it by dividing both sides by 2r:
(dr/dt) = 22 / (2r)
(dr/dt) = 11 / r
At the point (1, 15), the distance from the particle to the origin can be found by substituting x = 1 and y = 15 into the equation:
r^2 = x^2 + y^2
r^2 = 1^2 + 15^2
r^2 = 1 + 225
r^2 = 226
r = sqrt(226)
Substituting r = sqrt(226) into the equation (dr/dt) = 11 / r, we have:
(dr/dt) = 11 / sqrt(226)
Therefore, the rate of change of the distance from the particle to the origin at the instant the particle passes through the point (1,15) is 11 / sqrt(226) units per second.