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 4 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 find the equation for the distance between the particle and the origin as a function of x.
The distance between two points (x₁, y₁) and (x₂, y₂) in 2D space is given by the formula:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
In this case, the particle is moving along the curve y = 2√(3x + 7), and the origin is at (0, 0). So we have:
x₂ = x (the x-coordinate of the particle)
y₂ = 2√(3x + 7) (the y-coordinate of the particle)
x₁ = 0 (the x-coordinate of the origin)
y₁ = 0 (the y-coordinate of the origin)
Substituting these values into the distance formula, we get:
d = √((x - 0)² + (2√(3x + 7) - 0)²)
= √(x² + (2√(3x + 7))²)
= √(x² + 4(3x + 7))
= √(x² + 12x + 28)
Now, to find the rate of change of the distance from the particle to the origin, we need to differentiate this equation with respect to time (t):
d/dt(d) = d/dt(√(x² + 12x + 28))
Using the chain rule, we can simplify this:
d/dt(d) = [d/du(√u)] * [du/dx(x² + 12x + 28)] * dx/dt
where u = x² + 12x + 28
Taking the partial derivatives:
d/dt(d) = (1/(2√u)) * (2x + 12) * dx/dt
Now we need to find dx/dt from the information given in the problem. It is given that the x-coordinate of the particle increases at a rate of 4 units per second:
dx/dt = 4
Substituting this value back into the equation:
d/dt(d) = (1/(2√u)) * (2x + 12) * 4
Now we need to find the value of x and u at the given point (3, 8). Plugging in these values:
x = 3
u = 3² + 12(3) + 28 = 49
Substituting these values into the equation:
d/dt(d) = (1/(2√49)) * (2(3) + 12) * 4
= (1/14) * 30 * 4
= 2.143 units per second
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 approximately 2.143 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 between two points (x₁, y₁) and (x₂, y₂) is given by:
Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
In this case, we are interested in the distance between the particle and the origin, which is located at the point (0, 0). So, the distance formula becomes:
Distance = √((x - 0)² + (y - 0)²)
= √(x² + y²)
Now, we need to find the rate of change of this distance with respect to time. To do that, we need to take the derivative of the distance formula with respect to time (t). Using the chain rule, we have:
d(Distance)/dt = d(√(x² + y²))/dt
= (1/2)(x² + y²)^(-1/2) * d(x² + y²)/dt
To find d(x² + y²)/dt, we can use the given information that the x-coordinate is increasing at a rate of 4 units per second. This means dx/dt = 4 units per second.
Now, we need to find dy/dt, the rate of change of y with respect to time. We can find this using implicit differentiation. Differentiating y = 2 √(3x + 7) with respect to time, we get:
dy/dt = d(2 √(3x + 7))/dt
= 2 * d(√(3x + 7))/dt
= 2 * (1/2)(3x + 7)^(-1/2) * d(3x + 7)/dt
= (3x + 7)^(-1/2) * (3 * dx/dt)
Substituting dx/dt = 4, we have:
dy/dt = (3x + 7)^(-1/2) * 3 * 4
= 12 (3x + 7)^(-1/2)
Now, we have both dx/dt and dy/dt, so we can substitute them back into the expression for d(Distance)/dt:
d(Distance)/dt = (1/2)(x² + y²)^(-1/2) * d(x² + y²)/dt
= (1/2)(x² + y²)^(-1/2) * (2x * dx/dt + 2y * dy/dt)
= (x * dx/dt + y * dy/dt) / √(x² + y²)
Substituting the values x = 3, y = 8, dx/dt = 4, and dy/dt = 12 (3x + 7)^(-1/2):
d(Distance)/dt = (3 * 4 + 8 * 12(3*3 + 7)^(-1/2)) / √(3² + 8²)
Simplifying, we get:
d(Distance)/dt = (12 + 8 * 12 * 12(3*3 + 7)^(-1/2)) / √(3² + 8²)
Calculate the value of (3*3 + 7)^(-1/2) first:
(3*3 + 7)^(-1/2) = (9 + 7)^(-1/2)
= 16^(-1/2)
= 1/4
Substitute this value back into the expression:
d(Distance)/dt = (12 + 8 * 12 * 12 * (1/4)) / √(3² + 8²)
= (12 + 8 * 12 * 12 * 1/4) / √(9 + 64)
= (12 + 8 * 144/4) / √73
= (12 + 8 * 36) / √73
= (12 + 288) / √73
= 300 / √73
Therefore, the rate of change of the distance from the particle to the origin at the instant when it passes through the point (3, 8) is 300/√73 units per second.