To find the rate of change of the distance from the particle to the origin, we need to find the equation of the distance between the particle and the origin.
The distance formula between two points, (x1, y1) and (x2, y2), is given by:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, we want to find the distance between the particle and the origin, so (x1, y1) is (0, 0), and (x2, y2) is (x, y) where y = 3โ3x + 1.
Therefore, the equation of the distance between the particle and the origin is:
Distance = sqrt((x - 0)^2 + (3โ3x + 1 - 0)^2)
= sqrt(x^2 + (3โ3x + 1)^2)
To find the rate of change of the distance, we need to differentiate the distance equation with respect to time, t, and then substitute the given values.
Differentiating the distance equation with respect to t, we get:
d(Distance)/dt = d(sqrt(x^2 + (3โ3x + 1)^2))/dt
Using the chain rule and differentiating each term, we have:
d(Distance)/dt = (1/2)*(x^2 + (3โ3x + 1)^2)^(-1/2)*(2x*dx/dt) + (1/2)*(x^2 + (3โ3x + 1)^2)^(-1/2)*2(3โ3x + 1)*(9โ3*dx/dt)
Simplifying, we have:
d(Distance)/dt = (x*dx/dt + (3โ3x + 1)*(9โ3*dx/dt))/(sqrt(x^2 + (3โ3x + 1)^2))
Substituting the given values, x = 5 and dx/dt = 3, we get:
d(Distance)/dt = (5*3 + (3โ3*5 + 1)*(9โ3*3))/(sqrt(5^2 + (3โ3*5 + 1)^2))
= (15 + (45โ3 + 1)*27โ3)/(sqrt(25 + (45โ3 + 1)^2))
Calculating the values, we have:
d(Distance)/dt โ 56.57 units per second
Therefore, the rate of change of the distance from the particle to the origin at this instant is approximately 56.57 units per second.