#### To determine

**To find:** The quantities in the given problem.

#### Answer

The quantities in the given problem is dxdt=500 mi/h.

#### Explanation

**Given:**

A plane flying horizontally at an altitude of 1 mi and a speed of 500 mi/h passes directly over a radar station.

**Calculation:**

If let t be time (in hours) and x be the horizontal distance traveled by the plane (in mi), then we are given that dxdt=500 mi/h.

#### To determine

**To find:** The unknown.

#### Explanation

The rate at which the distance from the plane to the station is increase when it is 2 mi from the station. Let y be the distance from the plane to the station then find dydt when y=2 mi.

#### To determine

**To draw:** The picture of the plane position at time *t*.

#### Explanation

Draw the picture of the plane position at time *t* is given below.

From Figure 1, it is observed that the right angle triangle.

#### To determine

**To write:** The equation that relates the quantities.

#### Answer

The Equation is dydt=xy⋅dxdt.

#### Explanation

By the Pythagorean Theorem, y2=x2+1.

Here, the horizontal distance x increases as time t increases.

Therefore, the distance y also increases as time t increases.

Hence x and y both are the function of the time t.

Obtain dydt when y=2 mi and the speed of the plane is dxdt=500 mi/h.

Calculate the value of x when y=2 mi using y2=x2+1.

Substitute y=2 in y2=x2+1.

22=x2+14=x2+1x2=4−1x=±3

Since distance cannot be negative, x=3.

Differentiate the function y2=x2+1 with respect to the time t.

ddt(y2)=ddt(x2+1)2ydydt=2xdxdtdydt=xy⋅dxdt

#### To determine

**To find:** The rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station.

#### Answer

The required rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station is dydt=2503 mi/h ≈433 mi/h.

#### Explanation

Substitute 2 for y and 3 for x and 500 for dxdt in dydt.

dydt=32(500)=2505 mi/h ≈433 mi/h

Therefore, the required rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station is dydt=2503 mi/h ≈433 mi/h.