Question

This is more of a calculus question if this is out of your expertise then I understand

Answer 1

We are asked to determine the rate of change at which the distance "z" is decreasing.

To do that we will use the Pythagorean theorem, like this:

`z^2=x^2+y^2`

Now, we derivate both sides with respect to time. We will use implicit derivative and the chain rule:

`2z(dz)/(dt)=2x(dx)/(dt)+2y(dy)/(dt)`

Now, we can cancel out the 2:

`z(dz)/(dt)=x(dx)/(dt)+y(dy)/(dt)`

Now, we divide both sides by "z":

`(dz)/(dt)=(1)/(z)(x(dx)/(dt)+y(dy)/(dt))`

From the Pythagorean theorem we have that the value of "z" is:

`z=√(x^2+y^2)`

Substituting the value we get:

`(dz)/(dt)=(1)/(√(x^2+y^2))(x(dx)/(dt)+y(dy)/(dt))`

The derivative of "x" with respect to time is the velocity of car A and the derivative of "y" with respect to time is the velocity of car B. The velocities are negative since the values of "x" and "y" are decreasing. Substituting we get:

`(dz)/(dt)=(1)/(√((4miles)^2+(3miles)^2))((4miles)(-30(miles)/(hour))+(3miles)(-40(miles)/(hour)))`

Solving the operations:

`(dz)/(dt)=-48\frac{miles\text{ }}{hour}`

Therefore, the distances between the cars is decreasing at a rate of -48 miles/hour.