Final answer:
The rate of change of the distance from the particle to the origin at the point (1,15) is 2/3 units per second.
Step-by-step explanation:
To find the rate of change of the distance from the particle to the origin, we need to first find the distance function.
The distance from a point (x, y) to the origin can be found using the Pythagorean theorem, which states that the distance is equal to the square root of the sum of the squares of the x and y coordinates.
In this case, the distance function can be written as D(x) = sqrt(x^2 + y^2).
Next, we need to find the derivative of the distance function with respect to time. Since the x-coordinate is changing with a rate of 2 units per second, we can use the chain rule to obtain dD/dt = dD/dx * dx/dt.
The derivative of the distance function with respect to x is given by dD/dx = (1/2)*(2x/sqrt(4x+5)) = x/sqrt(4x+5).
The derivative of the x-coordinate with respect to time is dx/dt = 2 units per second.
Finally, we can substitute the given values into the expression for dD/dt to find the rate of change of the distance from the particle to the origin at the instant when it passes through the point (1,15).
We have dD/dt = (1/sqrt(4(1)+5)) * 2
= 2/sqrt(9)
= 2/3 units per second.