A particle is moving along the curve y=5sqrt(3x+1). As the particle passes through the point (5,20) its x-coordinate increases at a rate of 3 units per second. Find the rate of change of the distance from the particle to the origin at this instant.
dx/dt = 3 units/s at the time of interest
dy/dt = dy/dx * dx/dt = 3 dy/dx
dy/dx = (5/2)/sqrt(3x+1) = (5/2)/4 = 5/8
Compute dy/dt and then use
dR/dt = sqrt[(dy/dt)^2 + (dx/dt)^2] ]
sqrt[3^2 + (15/8)^2] = 3.538
I just plugged it in and it says it isn't right. =/
drwls left out the factor of 3 in the derivative, it should have been
dy/dx = (5/2)(3) / √(3x+1)
so when x=5
dy/dx = (15/2) / √16 = 15/8
take it from there
Thanks for the correction, Reiny :-)
I am getting sloppier in my old age
at the point .
To find the rate of change of the distance from the particle to the origin at the given instant, we need to first find an expression for the distance between the particle and the origin.
The distance between two points in a coordinate system can be found using the distance formula:
Distance = √[ (x2 - x1)^2 + (y2 - y1)^2 ]
In this case, the coordinates of the particle can be represented by (x, y) and the origin by (0, 0). So, we have:
Distance = √[ (x - 0)^2 + (y - 0)^2 ] = √(x^2 + y^2)
We are given the equation of the curve as y = 5√(3x + 1). To express the distance in terms of x, we substitute this value of y into the distance formula:
Distance = √[ x^2 + (5√(3x + 1))^2 ] = √[ x^2 + 25(3x + 1) ] = √(x^2 + 75x + 25)
Now, we want to find the rate of change of the distance from the particle to the origin, with respect to time (t). We know that the x-coordinate of the particle increases at a rate of 3 units per second. This tells us that dx/dt (rate of change of x with respect to t) is equal to 3.
To find the rate of change of the distance with respect to time, we use the chain rule from calculus. The chain rule states that if a function f(x) can be expressed as a composition of two functions, f(g(x)), then the derivative of f with respect to x is given by:
df/dx = df/dg * dg/dx
In our case, the distance function is expressed as a composition of two functions, f(x) = √(x^2 + 75x + 25), where g(x) = x^2 + 75x + 25. We want to find d(distance)/dt (the rate of change of the distance with respect to t), so we apply the chain rule:
d(distance)/dt = d(distance)/dx * dx/dt
We already know that dx/dt = 3. To find d(distance)/dx, we differentiate the distance function with respect to x:
d(distance)/dx = (1/2)(x^2 + 75x + 25)^(-1/2) * (2x + 75)
Simplifying this expression, we have:
d(distance)/dx = (x + 37.5) / √(x^2 + 75x + 25)
Now, we can substitute the values into the chain rule formula:
d(distance)/dt = (x + 37.5) / √(x^2 + 75x + 25) * dx/dt
Substituting the known values, dx/dt = 3, and the given x-coordinate of the particle at the instant is x = 5, we have:
d(distance)/dt = (5 + 37.5) / √(5^2 + 75(5) + 25) * 3
d(distance)/dt = 1.225 units per second
Therefore, the rate of change of the distance from the particle to the origin at this instant is approximately 1.225 units per second.