A particle is moving along the curve y=4sqrt(4x+1) . As the particle passes through the point (2,12), 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.
Thanks !!!
To find the rate of change of the distance from the particle to the origin, we will use the distance formula in two dimensions:
Distance = sqrt((x - x₀)² + (y - y₀)²),
where (x₀, y₀) represents the coordinates of the origin. In this case, the origin is (0, 0).
Now, let's differentiate both sides of the equation with respect to time (t) using the chain rule:
d(Distance)/dt = d(sqrt((x - x₀)² + (y - y₀)²))/dt
To simplify the equation, we'll find the derivatives of each part. First, we need to differentiate (x - x₀)² + (y - y₀)² with respect to t.
d((x - x₀)² + (y - y₀)²)/dt = 2(x - x₀)(dx/dt) + 2(y - y₀)(dy/dt)
Since the particle is moving along the curve y = 4√(4x + 1), we need to find dy/dx to substitute in the equation.
dy/dx = d(4√(4x + 1))/dx = 2(4x + 1)^(-1/2) * d(4x + 1)/dx = 2(4x + 1)^(-1/2) * 4 = 8/(4sqrt(4x + 1)).
Now, we can substitute dx/dt = 2 (given in the problem) and dy/dx into the equation:
d(Distance)/dt = 2(x - x₀) * 2 + 2(y - y₀) * (8/(4sqrt(4x + 1)))
Since we are interested in finding the rate of change of the distance from the particle to the origin when the particle passes through the point (2, 12), we substitute these values into the equation.
x = 2, x₀ = 0, y = 12, y₀ = 0:
d(Distance)/dt = 2(2 - 0) * 2 + 2(12 - 0) * (8/(4sqrt(4 * 2 + 1)))
Simplifying:
d(Distance)/dt = 4 * 2 + 2 * 12 * (8/(4sqrt(9)))
= 8 + 2 * 12 * (8/3)
= 8 + 16 * (8/3)
= 8 + 128/3
= 8 + 42.67
= 50.67 units per second.
Therefore, the rate of change of the distance from the particle to the origin at this instant is 50.67 units per second.
y= 4sqrt(4x+1)
dy/dt= 4*1/2*1/sqrt(4x+1)*4dx/dt
you are given dx/dt, solve for dy/dt
now, since r=sqrt(x^2+y^2)
dr/dt= 1/2 *1/sqrt(x^2+y^2)* (2xdx/dt+2ydy/dt)
solve for dr/dt