"Two cars start moving from the same point. One travels south at 48 mi/h and the other travels west at 20 mi/h. At what rate is the distance between the cars increasing four hours later?"

The direction of movement of both cars forms a right angle triangle. The distance travelled due south and due east by both cars represents the legs of the triangle. Their distance apart after t hours represents the hypotenuse of the right angle triangle.

Let x represent the length the shorter leg(west) of the right angle triangle.

Let y represent the length the longer leg(south) of the right angle triangle.

Let z represent the hypotenuse.

Applying Pythagoras theorem

Hypotenuse² = opposite side² + adjacent side²

Therefore

z² = x² + y²

To determine the rate at which the distances are changing, we would differentiate with respect to t. It becomes

2zdz/dt = 2xdx/dt + 2ydy/dt- - - -- - -1

One travels south at 48 mi/h and the other travels west at 20 mi/h. It means that

dx/dt = 20

dy/dt = 48

Distance = speed × time

Since t = 4 hours, then

x = 20 × 4 = 80 miles

y = 48 × 4 = 192 miles

z² = 80² + 192² = 6400 + 36864

z = √43264

z = 208 miles

Substituting these values into equation 1, it becomes

2 × 208 × dz/dt = 2 × 80 × 20 + 2 × 192 × 48

416dz/dt = 3200 + 18432

416dz/dt = 21632

dz/dt = 21632/416

dz/dt = 52 mph

ajeigbeibraheem ajeigbeibraheem · Answer 2 · 2021-10-29T11:53:25+02:00

Two cars start moving from the same point. One travels south at 48 mi/h and the other travels west at 20 mi/h. The rate at which the distance between the cars is increasing after 4 hours is 52 mi/h.

From the given information;

we are being told that the time (t) of both directions are the same, which = 4 hours later from their starting point.
Assuming the distance is constant since they are both the same.

Then, we can use the Pythagoras rule to attempt the question.

i.e.

(hypotenuse)² = (opposite)² + (adjacent)²

From the attached image below,

the opposite direction is due south = 48 mi/h
the adjacent direction is due west = 20 mi/h

Using the Pythagoras rule, we have:

(hypotenuse)² = 48² + 20²

(hypotenuse)² = 2304 mi/h + 400mi/h

(hypotenuse)² = 2704 mi/h

hypotenuse = [tex]\mathbf{\sqrt{2704 \ mi/h}}[/tex]

hypotenuse = 52 mi/h

Therefore, the distance between the cars increasing after four hours is the hypotenuse which is 52 mi/h.

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