Question

SCALCET8 3.9.017.MI. 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 two hours later

Answer 1

Answer:

The rate at which the distance between the cars increasing two hours later=52mi/h

Step-by-step explanation:

Let

Speed of one car, x'=48 mi/h

Speed of other car, y'=20 mi/h

We have to find the rate at which the distance between the cars increasing two hours later.

After 2 hours,

Distance traveled by one car

$x=48\times 2=96 mi$

Using the formula

$Distance=Time\times speed$

Distance traveled by other car

$y=20\times 2=40 mi$

Let z be the distance between two cars after 2 hours later

$z=\sqrt{x^2+y^2}$

Substitute the values

$z=\sqrt{(96)^2+(40)^2}$

z=104 mi

Now,

$z^2=x^2+y^2$

Differentiate w.r.t t

$2z\frac{dz}{dt}=2x\frac{dx}{dt}+2y\frac{dy}{dt}$

$z\frac{dz}{dt}=x\frac{dx}{dt}+y\frac{dy}{dt}$

Substitute the values

$104\frac{dz}{dt}=96\times 48+40\times 20$

$\frac{dz}{dt}=\frac{96\times 48+40\times 20}{104}$

$\frac{dz}{dt}=52mi/h$

Hence, the rate at which the distance between the cars increasing two hours later=52mi/h