Maths - Related rates.?

Question

Two straight roads intersect at right angles. Car A, moving along one of the roads, approaches
the intersection at 25km/h and car B, moving on the other road, approaches the intersection
at 30km/h. At what rate is the distance between the two cars changing when A is 0.5km from
the intersection and B is 1.2 km from the intersection?

I have formed two equations:
x= 0.5 +25t and y= 1.2 + 30t
- im stuck here and I dont know how to continue.

Thanks in advance!

Answer 1

Assume the following:

a = the distance of car A from the intersection
b = the distance of car B from the intersection
a' = a/t = the velocity of car A
b' = a/t = the velocity of car B
d = the distance between car A and car B

Using the Pythagorean theorem:

d^2 = a^2 + b^2

Taking the derivative:

(d/t)^2 = (a/t)^2 + (b/t)^2

Substituting d' = d/t, a' = a/t and b' = b/t:

d * d' = a * a' + b * b'

Substituting knowns from the problem statement:

d * d' = 0.5 * 25 + 1.2 * 30
d * d' = 48.5

We can determine d using the Pythagorean theorem:

d^2 = (0.5)^2 + (1.2)^2
d^2 = 0.25 + 1.44
d = 1.3

And:

1.3 * d' = 48.5
d' = 37.3 km/h

Answer 2

I'm going to change your equations a little. Since the cars are approaching the intersection, the distance is decreasing.

x = 0.5 - 25t
y = 1.2 - 30t

dx/dt = -25 km/hr
dy/dt = -30 km/hr

D = distance between the two cars
D² = x² + y²

At t = 0
D² = x² + y² = 0.25 + 1.44 = 1.69
D = 1.3 km

Take the derivative to find the critical values.

D² = x² + y²

2D(dD/dt) = 2d(dx/dt) + 2y(dy/dt)
D(dD/dt) = x(dx/dt) + y(dy/dt)

dD/dt = [x(dx/dt) + y(dy/dt)] / D

dD/dt = [0.5(-25) + 1.2(-30)] / 1.69 = -48.5/ 1.3

dD/dt = -485/13 ≈ -37.3 km / hr

Answer 3

First let me find the Distance when A is at 0.5 km and B is at 1.2 km.
(distance)^2 = A^2 + B^2
(distance)^2 = (0.5)^2 + (1.2)^2
distance b/w the cars = 1.3 km

Lets take the Pythagoras equation again and differentiate to get the rate of change of the distance,

(D)^2 = A^2 + B^2
Differentiate on both sides, we know that dA/dt = 25 km/hr and dB/dt = 30 km/hr. we need to find dD/dt.

2D(dD/dt) = 2A(dA/dt) + 2B(dB/dt)
2(1.3)(dD/dt) = 2(0.5)(25) + 2(1.2)(30)
2.6(dD/dt) = 25 + 72
dD/dt = 37.31 km/hr

Distance b/w them is changing at the rate of 31.31 km/hr.