A particle is moving along the curve . As the particle passes through the point , its -coordinate increases at a rate of units per second. Find the rate of change of the distance from the particle to the origin at this instant.
Gravel is being dumped from a conveyor belt at a rate of cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always equal to each other. How fast is the height of the pile increasing when the pile is feet high?
Hint: The volume of a right circular cone with height and radius of the base is given by .
DNE
To find the rate of change of the distance from the particle to the origin, we need to use the concept of the distance formula and differentiation.
Let's assume that the particle's position at time t is given by the coordinates (x(t), y(t)).
The distance between the particle's position and the origin can be calculated using the distance formula:
d(t) = sqrt(x(t)^2 + y(t)^2)
Now, we are given that the particle's x-coordinate is increasing at a rate of b units per second. This means that dx/dt = b.
As the particle is moving along the curve, its y-coordinate may also be changing, but we are not given any information about its rate of change.
To find the rate of change of the distance from the particle to the origin, we need to differentiate the distance function with respect to time:
(d/dt) d(t) = (d/dt) sqrt(x(t)^2 + y(t)^2)
Using the chain rule, the derivatives of x(t)^2 and y(t)^2 with respect to t can be calculated as:
(d/dt) x(t)^2 = 2x(t) * (dx/dt) = 2x(t) * b
(d/dt) y(t)^2 = 2y(t) * (dy/dt)
Since we are not given any information about dy/dt, we cannot calculate its value directly. However, we can make an observation that the particle is moving along the curve, and as it passes through the point, its x-coordinate is changing at a constant rate. This means that dy/dt must be zero.
Taking the derivative of the distance function with respect to time, we get:
(d/dt) d(t) = (d/dt) sqrt(x(t)^2 + y(t)^2)
= (1/2) * (x(t)^2 + y(t)^2)^(-1/2) * (2x(t) * b + 2y(t) * (dy/dt))
= (1/2) * (x(t)^2 + y(t)^2)^(-1/2) * 2x(t) * b
Now, we can substitute x(t) and b into the derivative expression to find the rate of change of the distance from the particle to the origin at that instant.
It's important to note that without knowing any information about dy/dt, we cannot determine the exact rate of change of the distance from the particle to the origin at that specific instant.