A particle moves in the plane along the curve y=xln(x). At what rate is the distance from the origin increasing at the moment its x-coordinate is 2 cm and its x-coordinate is increasing at a rate of 17 cm/sec?
distance=sqrt(y^2 + x^2)
ddistance/dt= 1/2distance *( 2ydy/dt+2xdx/dt)
now examine y dy/dt= x ln(x) (ln(x)+1) dx/dt
ddistance/dt=rateyou want=1/2distance*(2xlnx+1)17cm/sec + x dx/dt)
check that, I typed it in a hurry.
To find the rate at which the distance from the origin is increasing, we need to find the derivative of the distance function with respect to time. Let's break down the problem into steps:
1. Find the distance function from the origin:
The distance between a point (x, y) and the origin (0, 0) is given by the distance formula:
D = sqrt(x^2 + y^2)
2. Express y in terms of x:
Given y = xln(x), we can substitute this expression for y in the distance formula:
D = sqrt(x^2 + (xln(x))^2)
3. Take the derivative of the distance function with respect to time:
Now that we have the distance function in terms of x, we can differentiate it with respect to time using the chain rule:
dD/dt = dD/dx * dx/dt
4. Find dx/dt:
Given that the x-coordinate is increasing at a rate of 17 cm/sec, we have dx/dt = 17 cm/sec.
5. Differentiate the distance function with respect to x:
To find dD/dx, we differentiate the distance function (D) with respect to x:
dD/dx = (1/2) * (x^2 + (xln(x))^2)^(-1/2) * (2x + 2xln(x) + x^2/x)
6. Evaluate the derivative at the given x-coordinate:
Substitute x = 2 into the derivative expression obtained in step 5:
dD/dx = (1/2) * (2^2 + (2ln(2))^2)^(-1/2) * (2*2 + 2*2ln(2) + 2^2/2)
= (1/2) * (4 + (2ln(2))^2)^(-1/2) * (4 + 4ln(2) + 2)
= (1/2) * (4 + 4ln(2) + 2)^(-1/2) * (10 + 4ln(2))
7. Evaluate the rate at which x is changing at the given x-coordinate:
Substitute dx/dt = 17 into the derivative expression obtained in step 7:
dD/dt = (1/2) * (4 + 4ln(2) + 2)^(-1/2) * (10 + 4ln(2)) * (17)
8. Simplify:
Evaluate the expression obtained in step 7 to get the rate at which the distance from the origin is increasing at the given moment.