A particle is moving along the curve y = 2 √{3 x + 7}. 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.
To find the rate of change of the distance from the particle to the origin, we need to determine the distance function first.
The distance between two points in a plane can be found using the distance formula:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
In this case, the two points are the particle's current position (x, y) and the origin (0, 0).
Since the particle is moving along the curve y = 2√(3x + 7), we can substitute this y-value into the distance formula:
d = √((x - 0)² + (2√(3x + 7) - 0)²)
= √(x² + 4(3x + 7))
Now, we differentiate both sides of the equation with respect to time to find the rate of change of the distance:
d/dt [d] = d/dt [√(x² + 4(3x + 7))]
Using the chain rule, we can simplify this expression:
d/dt [d] = 1/2(x² + 4(3x + 7))^(-1/2) * d/dt [x² + 4(3x + 7)]
Next, we substitute the given information:
dx/dt = 5 (the rate at which the particle's x-coordinate is increasing).
Now, we can find the rate of change of the distance by plugging in the values:
d/dt [d] = 1/2(3² + 4(3(3) + 7))^(-1/2) * 5² + 4(3(5) + 7))]
Simplifying further, we have:
d/dt [d] = 1/2(9 + 4(24))^(-1/2) * (25 + 4(22))
= 1/2(9 + 96)^(-1/2) * (25 + 88)
= 1/2(105)^(-1/2) * 113
= 1/(2√105) * 113
= 113/(2√105)
Therefore, the rate of change of the distance from the particle to the origin at this instant is 113/(2√105) units per second.
To find the rate of change of the distance from the particle to the origin, we need to use the distance formula.
The distance formula in two dimensions is given by:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
In this case, the point (x₁, y₁) is the origin (0, 0) and the point (x₂, y₂) is the position of the particle on the curve. We are given that the particle passes through the point (3, 8), so the coordinates are (3, 8).
Substituting these values into the distance formula, we have:
d = √((3 - 0)² + (8 - 0)²)
d = √(3² + 8²)
d = √(9 + 64)
d = √73
So the distance from the particle to the origin is √73.
To find the rate of change of this distance, we need to differentiate the distance formula with respect to time. Since x is changing with respect to time, we can use the chain rule for differentiation.
Differentiating both sides of the distance formula with respect to time, we get:
d/ dt = (1/2√73) * [(2 * (3 - 0) * dx/dt) + (2 * (8 - 0) * dy/dt)]
Given that dx/dt = 5 (since the x-coordinate is increasing at a rate of 5 units per second) and dy/dt is the rate of change of y with respect to time, we need to find dy/dt.
To find dy/dt, we can differentiate the equation y = 2 √(3x + 7) with respect to time:
dy/dt = d/dt (2 √(3x + 7))
Using the chain rule, we have:
dy/dt = 2 * (1/2√(3x + 7)) * d/dt (3x + 7)
dy/dt = (1/√(3x + 7)) * d/dt (3x + 7)
The derivative of (3x + 7) with respect to time is simply 3 dx/dt (since the derivative of a constant term is zero).
dy/dt = (1/√(3x + 7)) * 3 dx/dt
Substituting dx/dt = 5 and the coordinates of the particle (x = 3), we have:
dy/dt = (1/√(3(3) + 7)) * 3 * 5
dy/dt = (1/√(9 + 7)) * 15
dy/dt = (1/√16) * 15
dy/dt = (1/4) * 15
dy/dt = 15/4
So the rate of change of y with respect to time is 15/4.
Now we have all the values needed to find d/dt, the rate of change of the distance from the particle to the origin:
d/ dt = (1/2√73) * [(2 * (3 - 0) * 5) + (2 * (8 - 0) * 15/4)]
d/ dt = (1/2√73) * (6 * 5 + 16 * 15/4)
d/ dt = (1/2√73) * (30 + 60)
d/ dt = (1/2√73) * 90
Simplifying, we get:
d/ dt = 45/√73
Therefore, 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) is 45/√73 units per second.