Question

Two airplanes are flying in the air at the same height. Airplane A is flying east at 300mi/h and airplane B is flying north at 200mi/h. If they are both heading to the same airport, located 90 miles east of airplane A and 120 miles north of airplane B, at what rate is the distance between the airplanes changing

Answer 1

Answer:

$\frac{dAB}{dt}=340 mil/h$

Step-by-step explanation:

The change of distance over time of the plain A is 300 mi/hour and 200 mi/hour for plane B. O is the point of the airport.

So, the distance from A to O AO = 90 miles and BO = 120 miles.

Now, we have a right triangle here.  We can use the Pythagorean theorem, so the distance between the planes will be:

$AB^{2} =AO^{2}+BO^{2}$ (1)

$AB =\sqrt{AO^{2}+BO^{2}}=$

$AB =\sqrt{90^{2}+120^{2}}=150 miles$

If we take the derivative of the equation (1) we could find the change of the distance between planes.

$2AB\frac{dAB}{dt}=2AO\frac{dAO}{dt}+2BO\frac{dBO}{dt}$

$2*150\frac{dAB}{dt}=2*90*300+2*120*200=102000 mil/h$

$\frac{dAB}{dt}=\frac{102000}{150*2}$

Finally,

$\frac{dAB}{dt}=340 mil/h$

I hope it helps you!

Answer 2

The rate at which the distance between the airplanes is changing is;

dC/dt = 340 mi/h

We are given;

Speed of Airplane A; dA/dt = 300 mi/h

Speed of Airplane B; dB/dt = 200 mi/h

Since airplane A is flying east and Airplane B is flying north, it means they will form a right angle triangle and so, we can use Pythagoras theorem to find the distance between them.

Let the distance between them be C. Thus;

C² = A² + B² --- eq 1

Differentiating with respect to t gives;

2C(dC/dt) = 2A(dA/dt) + 2B(dB/dt)

Divide through by 2 to get;

C(dC/dt) = A(dA/dt) + B(dB/dt)   ---eq 2

where dC/dt is the rate at which the distance between the airplanes is changing.

Now, let us find C from C² = A² + B²;

C = √(A² + B²)

Putting A = 90 miles and B = 120 miles;

C = √(90² + 120²)

C = √22500

C = 150 miles

Putting the relevant values into eq 2 gives;

150(dC/dt) = (90 × 300) + (120 × 200)

150(dC/dt) = 51000

dC/dt = 51000/150

dC/dt = 340 mi/h

Read more at; https://brainly.com/question/13893841