[CALCULUS] A DOG and A CAT need to separate at a spot. The dog walks south at a rate of 1feet per second. The cat walks east at a rate of 1.5 feet per second....

Question

A DOG and A CAT need to separate at a spot. The dog walks south at a rate of 1feet per second. The cat walks east at a rate of 1.5 feet per second. How fast are they walking away from each other after 1 minute?

Answer 1

Since the pets are moving away from each other at 90 degrees (one south, the other east), we can envision the problem by a right triangle. Let x = the eastward bound cat and y = the southward bound dog. Let z be the distance between them. By Pythagorean theorem x^2 + y^2 = z^2. Now differentiate with respect to time implicitly (d/dt) results in (2x)dx/dt + (2y)dy/dt = (2z)dz/dt. Divide all terms by 2

x*dx/dt + y*dy/dt = z*dz/dt.

At time i minute, the x = 60 feet, the y = 90 feet. The Z is then .106.16 Feet. dx/dt is given as 1, dy/dt is given as 1.5. Solve for dz/dt.

Answer 2

Let D(t) = the distance between the animals at time t

D(t) = sqrt(2.25t² + d²) = t*sqrt(3.25) (Pythagorean theorem!)

dD/dt=sqrt(3.25)