Question

A particle on the x-axis is moving to the right at 2 units per second. At a certain instant it is at the point (5,0). How rapidly it the distance between the particle and the point (0,9) on the y-axis changing at that instant

Answer 1

Answer:

dr/dt = 1.94 units per second

Explanation:

A particle is moving on the x-axis to the right at 2 u/s.

To know how is changing the distance of the particle respect to the point (0,9), on the y-axis, you first take into account that the distance between charge and a point over the y-axis is given by:

$r^2=x^2+y^2$          (1)

Next, you calculate implicitly the derivative of the equation (1) respect to t:

$\frac{d}{dt}r^2=\frac{d}{dt}[x^2+y^2]\\\\2r\frac{dr}{dt}=2x\frac{dx}{dt}+2y\frac{dy}{dt}$       (2)

Next, you solve the previous equation for dr/dx:

$\frac{dr}{dt}=\frac{x(dx/dt)+y(dy/dt)}{\sqrt{x^2+y^2}}$       (3)

dx/dt: the speed of the particle on the x-axis = 2

dy/dt: speed of the particle on the y-axis = 0

For the instant given in the statement, you have that:

x = 5

y = 9

Then, you replace the values for x, y, dx/dt and dy/dt in the equation (3):

$\frac{dr}{dt}=\frac{2(5)(2)+0}{\sqrt{(5)^2+(9)^2}}=1.94$

The speed of change of the distance between particle and point (0,9) is 1.94 units per second